Question 576·Medium·Algebra

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$5x - 15 = \dfrac{b(x - 3)}{2}$

In the equation above, $b$ is a constant. If the equation is true for all real numbers $x$ , what is the value of $b$ ?

The phrase true for all real numbers signals an identity: both sides must agree at every input, not just one. Factor first to expose any shared part, then use a Desmos regression at several inputs to find the missing constant. Avoid an input that makes the shared factor zero; it cannot tell you the constant.

Hints

Hint 1
Look for a shared factor on both sides. Factor $5$ out of $5x-15$ to write it using $(x-3)$ . What must be true of the numbers multiplying that factor if the equation works for every $x$ ?
Hint 2
At $x=3$ , the factor $(x-3)$ is zero, so both sides equal zero regardless of $b$ . To find $b$ , test inputs where that factor isn't zero.

Step-by-step

Approach 1: Factor, then fit the constant in Desmos
Step 1
Expose the shared factor
Factor $5$ from the left, since $5x-15=5(x-3)$ : $5(x-3)=\frac{b(x-3)}{2}$ . Both sides contain $(x-3)$ . At $x=3$ , both sides are $0$ no matter what $b$ is, so that input cannot determine it.
Step 2
Find the constant and check the identity
Type $x_1=[1,2,4,5]$ , then $5x_1-15\sim\frac{b(x_1-3)}{2}$ . The symbol $\sim$ tells Desmos to fit the unknown constant $b$ across those inputs. Under PARAMETERS, Desmos shows $b=10$ . Since $\frac{10(x-3)}{2}=5(x-3)$ , the sides match for every $x$ , not only the tested inputs. For an equation to hold for every input, its two sides must be the same expression. The value of $b$ is $10$ . Choice C.

Approach 2: Match the coefficients directly
Step 1
Match the multipliers
Instead of testing inputs, compare the coefficients, the numbers multiplying $(x-3)$ in the factored equation. Because the equation holds for every $x$ , those multipliers must agree: $5=\frac{b}{2}$ .
Step 2
Solve for the constant
Multiply both sides by $2$ : $10=b$ . Both sides of the original equation now equal $5(x-3)$ , so $b$ is $10$ . Choice C.

Question 576·Medium·Algebra

Strategy

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Step-by-step

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