Practice · No solution means 0 times x equals a nonzero number · Sensim Math

Variant 1 easy

There is no value of $x$ that satisfies the proportion $0.6(x-2):a=0.4(x+1):2$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.6(x-2):a=0.4(x+1):2$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.4(x+1)=0.6(x-2)\times 2 \;\Rightarrow\; 0.4a(x+1)=1.2(x-2)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

4a(x+1)=12(x-2) \;\Rightarrow\; 4ax+4a=12x-24 \;\Rightarrow\; (4a-12)x=-4a-24

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

4a-12\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

4a-12=0 \;\text{and}\; -4a-24\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=3

, and then

-4a-24=-36\neq 0

, so the condition holds.

a=3,\quad 0\times x=-36

The equation becomes impossible.

Answer:

3

4 · Reviewdoes it hold up?

With $a=3$ the proportion asks $0.6(x-2)\times 2=3\times 0.4(x+1)$ , that is, $1.2x-2.4=1.2x+1.2$ . Subtracting $1.2x$ leaves $-2.4=1.2$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.2x-2.4$ and $0.4a\,x+0.4a$ . They never meet only when they are parallel (same slope, $0.4a=1.2$ , so $a=3$ ) and not the same line ( $-2.4\neq 1.2$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=-36

.

Variant 2 easy

There is no value of $x$ that satisfies the proportion $0.7(x+1):a=0.1(x-1):2$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.7(x+1):a=0.1(x-1):2$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.1(x-1)=0.7(x+1)\times 2 \;\Rightarrow\; 0.1a(x-1)=1.4(x+1)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

a(x-1)=14(x+1) \;\Rightarrow\; ax-a=14x+14 \;\Rightarrow\; (a-14)x=a+14

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

a-14\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

a-14=0 \;\text{and}\; a+14\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=14

, and then

a+14=28\neq 0

, so the condition holds.

a=14,\quad 0\times x=28

The equation becomes impossible.

Answer:

14

4 · Reviewdoes it hold up?

With $a=14$ the proportion asks $0.7(x+1)\times 2=14\times 0.1(x-1)$ , that is, $1.4x+1.4=1.4x-1.4$ . Subtracting $1.4x$ leaves $1.4=-1.4$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.4x+1.4$ and $0.1a\,x-0.1a$ . They never meet only when they are parallel (same slope, $0.1a=1.4$ , so $a=14$ ) and not the same line ( $1.4\neq -1.4$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=28

.

Variant 3 easy

There is no value of $x$ that satisfies the proportion $0.9(x+2):a=0.3(x-1):2$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.9(x+2):a=0.3(x-1):2$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.3(x-1)=0.9(x+2)\times 2 \;\Rightarrow\; 0.3a(x-1)=1.8(x+2)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

3a(x-1)=18(x+2) \;\Rightarrow\; 3ax-3a=18x+36 \;\Rightarrow\; (3a-18)x=3a+36

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

3a-18\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

3a-18=0 \;\text{and}\; 3a+36\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=6

, and then

3a+36=54\neq 0

, so the condition holds.

a=6,\quad 0\times x=54

The equation becomes impossible.

Answer:

6

4 · Reviewdoes it hold up?

With $a=6$ the proportion asks $0.9(x+2)\times 2=6\times 0.3(x-1)$ , that is, $1.8x+3.6=1.8x-1.8$ . Subtracting $1.8x$ leaves $3.6=-1.8$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.8x+3.6$ and $0.3a\,x-0.3a$ . They never meet only when they are parallel (same slope, $0.3a=1.8$ , so $a=6$ ) and not the same line ( $3.6\neq -1.8$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=54

.

Variant 4 medium

There is no value of $x$ that satisfies the proportion $0.5(x-1):a=0.1(x+2):3$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.5(x-1):a=0.1(x+2):3$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.1(x+2)=0.5(x-1)\times 3 \;\Rightarrow\; 0.1a(x+2)=1.5(x-1)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

a(x+2)=15(x-1) \;\Rightarrow\; ax+2a=15x-15 \;\Rightarrow\; (a-15)x=-2a-15

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

a-15\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

a-15=0 \;\text{and}\; -2a-15\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=15

, and then

-2a-15=-45\neq 0

, so the condition holds.

a=15,\quad 0\times x=-45

The equation becomes impossible.

Answer:

15

4 · Reviewdoes it hold up?

With $a=15$ the proportion asks $0.5(x-1)\times 3=15\times 0.1(x+2)$ , that is, $1.5x-1.5=1.5x+3$ . Subtracting $1.5x$ leaves $-1.5=3$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.5x-1.5$ and $0.1a\,x+0.2a$ . They never meet only when they are parallel (same slope, $0.1a=1.5$ , so $a=15$ ) and not the same line ( $-1.5\neq 3$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=-45

.

Variant 5 easy

There is no value of $x$ that satisfies the proportion $0.4(x-1):a=0.2(x+3):3$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.4(x-1):a=0.2(x+3):3$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.2(x+3)=0.4(x-1)\times 3 \;\Rightarrow\; 0.2a(x+3)=1.2(x-1)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

2a(x+3)=12(x-1) \;\Rightarrow\; 2ax+6a=12x-12 \;\Rightarrow\; (2a-12)x=-6a-12

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

2a-12\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

2a-12=0 \;\text{and}\; -6a-12\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=6

, and then

-6a-12=-48\neq 0

, so the condition holds.

a=6,\quad 0\times x=-48

The equation becomes impossible.

Answer:

6

4 · Reviewdoes it hold up?

With $a=6$ the proportion asks $0.4(x-1)\times 3=6\times 0.2(x+3)$ , that is, $1.2x-1.2=1.2x+3.6$ . Subtracting $1.2x$ leaves $-1.2=3.6$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.2x-1.2$ and $0.2a\,x+0.6a$ . They never meet only when they are parallel (same slope, $0.2a=1.2$ , so $a=6$ ) and not the same line ( $-1.2\neq 3.6$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=-48

.

Variant 6 medium

There is no value of $x$ that satisfies the proportion $0.5(x+1):a=0.2(x-3):4$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.5(x+1):a=0.2(x-3):4$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.2(x-3)=0.5(x+1)\times 4 \;\Rightarrow\; 0.2a(x-3)=2(x+1)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

2a(x-3)=20(x+1) \;\Rightarrow\; 2ax-6a=20x+20 \;\Rightarrow\; (2a-20)x=6a+20

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

2a-20\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

2a-20=0 \;\text{and}\; 6a+20\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=10

, and then

6a+20=80\neq 0

, so the condition holds.

a=10,\quad 0\times x=80

The equation becomes impossible.

Answer:

10

4 · Reviewdoes it hold up?

With $a=10$ the proportion asks $0.5(x+1)\times 4=10\times 0.2(x-3)$ , that is, $2x+2=2x-6$ . Subtracting $2x$ leaves $2=-6$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $2x+2$ and $0.2a\,x-0.6a$ . They never meet only when they are parallel (same slope, $0.2a=2$ , so $a=10$ ) and not the same line ( $2\neq -6$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=80

.

Variant 7 medium

There is no value of $x$ that satisfies the proportion $0.2(x+1):a=0.1(x-2):4$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.2(x+1):a=0.1(x-2):4$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.1(x-2)=0.2(x+1)\times 4 \;\Rightarrow\; 0.1a(x-2)=0.8(x+1)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

a(x-2)=8(x+1) \;\Rightarrow\; ax-2a=8x+8 \;\Rightarrow\; (a-8)x=2a+8

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

a-8\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

a-8=0 \;\text{and}\; 2a+8\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=8

, and then

2a+8=24\neq 0

, so the condition holds.

a=8,\quad 0\times x=24

The equation becomes impossible.

Answer:

8

4 · Reviewdoes it hold up?

With $a=8$ the proportion asks $0.2(x+1)\times 4=8\times 0.1(x-2)$ , that is, $0.8x+0.8=0.8x-1.6$ . Subtracting $0.8x$ leaves $0.8=-1.6$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $0.8x+0.8$ and $0.1a\,x-0.2a$ . They never meet only when they are parallel (same slope, $0.1a=0.8$ , so $a=8$ ) and not the same line ( $0.8\neq -1.6$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=24

.

Variant 8 medium

There is no value of $x$ that satisfies the proportion $0.3(x+1):a=0.2(x+2):4$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.3(x+1):a=0.2(x+2):4$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.2(x+2)=0.3(x+1)\times 4 \;\Rightarrow\; 0.2a(x+2)=1.2(x+1)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

2a(x+2)=12(x+1) \;\Rightarrow\; 2ax+4a=12x+12 \;\Rightarrow\; (2a-12)x=-4a+12

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

2a-12\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

2a-12=0 \;\text{and}\; -4a+12\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=6

, and then

-4a+12=-12\neq 0

, so the condition holds.

a=6,\quad 0\times x=-12

The equation becomes impossible.

Answer:

6

4 · Reviewdoes it hold up?

With $a=6$ the proportion asks $0.3(x+1)\times 4=6\times 0.2(x+2)$ , that is, $1.2x+1.2=1.2x+2.4$ . Subtracting $1.2x$ leaves $1.2=2.4$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.2x+1.2$ and $0.2a\,x+0.4a$ . They never meet only when they are parallel (same slope, $0.2a=1.2$ , so $a=6$ ) and not the same line ( $1.2\neq 2.4$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=-12

.

Variant 9 hard

There is no value of $x$ that satisfies the proportion $0.2(x-3):a=0.5(x+1):5$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.2(x-3):a=0.5(x+1):5$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.5(x+1)=0.2(x-3)\times 5 \;\Rightarrow\; 0.5a(x+1)=1(x-3)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

5a(x+1)=10(x-3) \;\Rightarrow\; 5ax+5a=10x-30 \;\Rightarrow\; (5a-10)x=-5a-30

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

5a-10\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

5a-10=0 \;\text{and}\; -5a-30\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=2

, and then

-5a-30=-40\neq 0

, so the condition holds.

a=2,\quad 0\times x=-40

The equation becomes impossible.

Answer:

2

4 · Reviewdoes it hold up?

With $a=2$ the proportion asks $0.2(x-3)\times 5=2\times 0.5(x+1)$ , that is, $x-3=x+1$ . Subtracting $x$ leaves $-3=1$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $x-3$ and $0.5a\,x+0.5a$ . They never meet only when they are parallel (same slope, $0.5a=1$ , so $a=2$ ) and not the same line ( $-3\neq 1$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=-40

.

Variant 10 hard

There is no value of $x$ that satisfies the proportion $0.3(x+2):a=0.1(x-1):5$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.3(x+2):a=0.1(x-1):5$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.1(x-1)=0.3(x+2)\times 5 \;\Rightarrow\; 0.1a(x-1)=1.5(x+2)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

a(x-1)=15(x+2) \;\Rightarrow\; ax-a=15x+30 \;\Rightarrow\; (a-15)x=a+30

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

a-15\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

a-15=0 \;\text{and}\; a+30\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=15

, and then

a+30=45\neq 0

, so the condition holds.

a=15,\quad 0\times x=45

The equation becomes impossible.

Answer:

15

4 · Reviewdoes it hold up?

With $a=15$ the proportion asks $0.3(x+2)\times 5=15\times 0.1(x-1)$ , that is, $1.5x+3=1.5x-1.5$ . Subtracting $1.5x$ leaves $3=-1.5$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.5x+3$ and $0.1a\,x-0.1a$ . They never meet only when they are parallel (same slope, $0.1a=1.5$ , so $a=15$ ) and not the same line ( $3\neq -1.5$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=45

.

Variant 11 hard

There is no value of $x$ that satisfies the proportion $0.1(x+4):a=0.2(x-1):6$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.1(x+4):a=0.2(x-1):6$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.2(x-1)=0.1(x+4)\times 6 \;\Rightarrow\; 0.2a(x-1)=0.6(x+4)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

2a(x-1)=6(x+4) \;\Rightarrow\; 2ax-2a=6x+24 \;\Rightarrow\; (2a-6)x=2a+24

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

2a-6\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

2a-6=0 \;\text{and}\; 2a+24\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=3

, and then

2a+24=30\neq 0

, so the condition holds.

a=3,\quad 0\times x=30

The equation becomes impossible.

Answer:

3

4 · Reviewdoes it hold up?

With $a=3$ the proportion asks $0.1(x+4)\times 6=3\times 0.2(x-1)$ , that is, $0.6x+2.4=0.6x-0.6$ . Subtracting $0.6x$ leaves $2.4=-0.6$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $0.6x+2.4$ and $0.2a\,x-0.2a$ . They never meet only when they are parallel (same slope, $0.2a=0.6$ , so $a=3$ ) and not the same line ( $2.4\neq -0.6$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=30

.

Variant 12 hard

There is no value of $x$ that satisfies the proportion $0.2(x+3):a=0.3(x-1):6$ . Find the value of the constant $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Turn the proportion into an ordinary equation in $x$ . Then choose $a$ so that this equation has no solution at all.

Givens

The proportion $0.2(x+3):a=0.3(x-1):6$ .
No value of $x$ satisfies it.

Unknowns

The constant $a$ .

Constraints

An equation $px=q$ has no solution exactly when $p=0$ and $q\neq 0$ .

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #13 Convert to Algebra#3 Eliminate Possibilities

Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form $(\dots)x=(\dots)$ ; second, read off when such an equation is impossible.

3 · Execute4 carry out the plan

1Proportion to equation

#7 Identify Subproblems 7.RP.A.2

In

A:B=C:D

, the product of the means equals the product of the extremes:

B\times C=A\times D

.

a\times 0.3(x-1)=0.2(x+3)\times 6 \;\Rightarrow\; 0.3a(x-1)=1.2(x+3)

A proportion is two equal ratios.

2Clear decimals and collect

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

10

, expand, and gather the

x

terms.

3a(x-1)=12(x+3) \;\Rightarrow\; 3ax-3a=12x+36 \;\Rightarrow\; (3a-12)x=3a+36

Now it has the form

px=q

.

3When is there no solution?

#3 Eliminate Possibilities 8.EE.C.7

If

3a-12\neq 0

we could divide and get one solution. So the coefficient must be

0

, while the right side must not be

0

.

3a-12=0 \;\text{and}\; 3a+36\neq 0

0\times x

is always

0

, so it can never equal a nonzero number.

4Find a

#13 Convert to Algebra 8.EE.C.7

a=4

, and then

3a+36=48\neq 0

, so the condition holds.

a=4,\quad 0\times x=48

The equation becomes impossible.

Answer:

4

4 · Reviewdoes it hold up?

With $a=4$ the proportion asks $0.2(x+3)\times 6=4\times 0.3(x-1)$ , that is, $1.2x+3.6=1.2x-1.2$ . Subtracting $1.2x$ leaves $3.6=-1.2$ , which is false for every $x$ .

Another way: Think of the two sides as lines in $x$ : $1.2x+3.6$ and $0.3a\,x-0.3a$ . They never meet only when they are parallel (same slope, $0.3a=1.2$ , so $a=4$ ) and not the same line ( $3.6\neq -1.2$ ).

Standardsmin grade 8

8.EE.C.7 Solve linear equations in one variable — Deciding when a linear equation has no solution.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Turning the proportion into an equation using equal ratios.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Clearing decimals and rearranging into the form $px=q$.

💡Takeaway. "No solution" means the

x

's cancel out and leave a false statement like

0=48

.

No solution means 0 times x equals a nonzero number · 12 practice problems

Generated variants — 12

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