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Problem 232

Algebra Difficulty 1.9 Find the answer CEMC Cayley

If $a(x+b)=3 x+12$ for all values of $x$ , what is the value of $a+b$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since $a(x+b)=3 x+12$ for all $x$ , then $a x+a b=3 x+12$ for all $x$ . Since the equation is true for all $x$ , then the coefficients on the left side must match the coefficients on the right side. Therefore, $a=3$ and $a b=12$ , which gives $3 b=12$ or $b=4$ . Finally, $a+b=3+4=7$ .

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