Depth · one concept from first principles · step 2 of 4

A relation fixes a ratio, so the expression is constant

Suppose $a+4b=2(a-b)$ , where $ab\neq 0$ and $a\neq b$ . The value of $\frac{3a+2b}{a-b}$ is the same as the solution of the equation $\frac{x+21}{5}-ax=\frac{x-a}{3}$ in $x$ . Find the value of $-a^2+a-1$ .

Expressions

Your answer

How to solve

Strategy Identify Subproblems — Split it into three small problems: turn the relation into a=6b; use that to show the fraction always equals 4; then put x=4 into the second equation to get a.

1STEP 1

Simplify the relation

Expand and collect terms.

a+4b=2a-2b → 6b=a

2STEP 2

The fraction is a fixed number

Replace a with 6b; the b cancels because b ≠ 0.

(3a+2b)/(a-b)=(18b+2b)/(6b-b)=20b/(5b)=4

3STEP 3

Put x = 4 into the equation

The solution is x=4, so substitute it.

(4+21)/5-4a=(4-a)/3 → 5-4a=(4-a)/3

4STEP 4

Solve for a

Multiply both sides by 3 and collect.

15-12a=4-a → 11=11a → a=1

5STEP 5

Evaluate

Substitute a=1.

-1²+1-1=-1

Answer

-1

With a=1, b=1/6: a+4b=5/3 and 2(a-b)=2×5/6=5/3. In the equation with a=1: 25/5-4=1 and (4-1)/3=1, so x=4 is indeed the solution. Also ab ≠ 0 and a ≠ b.

Takeaway

When a relation ties two letters together, a fraction made of them can turn into a plain number.

Simplify the relation
The fraction is a fixed number
Put x = 4 into the equation
Solve for a
Evaluate

Where next?

Level upsuggested

Turn the condition into x+y in terms of xy Rewrite numerators using the given total

Another one like thissuggested

12 practice problems

Same look, different logicsuggested

Write the solution in a, then filter by wholeness Count the children two ways and equate

Step back and solidifysuggested

Sum pyramid: symmetric terms cancel at the top Reduce the investment ratio to whole numbers

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 12 problems