Depth · one concept from first principles · step 2 of 4
True for every k means an identity in k
The equation in always has as a solution, no matter what value takes. Find the sum of the constants and .
Expressions
Your answer
How to solve
Strategy Change Focus / Count the Complement — After substituting x=2, stop treating k as a fixed number and look at it as the variable: an equation that holds for every k is an identity in k, so both sides must have the same k-part and the same constant part.
1STEP 1
Substitute the solution
Since x=2 is always a solution, put x=2 into the equation and expand.
(2k+3) × 2-3a=bk+3 → 4k+6-3a=bk+3
Now only k, a, and b are left.
7.EE.A.1Convert To Algebra2STEP 2
Read it as an identity in k
The line 4k+6-3a=bk+3 is true for every k. Two expressions in k that agree for every k must have the same coefficient of k.
4k = bk for all k → b=4
If the k-parts differed, changing k would make the two sides drift apart.
8.EE.C.7Change Focus Count The ComplementBecause is true for every , must equal .
Why?
Two expressions in that agree for every value of must carry the same number of s.
🧱Equal polynomials match coefficient by coefficientTwo polynomials that agree for every input must have the same coefficient on every power, so one identity becomes a list of plain equations.
Why?
You can see it by moving from to : the left side rises by and the right side by , and two sides that were equal and stay equal must have risen by the same amount.
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
3STEP 3
Match the constant parts
With the k-parts equal, the parts without k must also be equal.
6-3a=3 → 3a=3 → a=1
A one-step equation finishes it.
7.EE.B.4Convert To Algebra4STEP 4
Add the constants
Add the two values.
a+b=1+4=5
Both constants are now known.
7.EE.B.4Convert To AlgebraAnswer
5
With a=1 and b=4 the equation is (2k+3)x-3=4k+3. At x=2 the left side is 4k+6-3=4k+3, equal to the right side for every k.
Takeaway
"True no matter what k is" is a big clue: match the k-parts and the plain-number parts separately.
- Substitute the solution
- Read it as an identity in k
- Match the constant parts
- Add the constants
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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