Question 312·Hard·Algebra

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The linear function $g$ is defined by $g(t)=kt+m$ , where $k$ and $m$ are constants. The function satisfies

g(3t-5)=2t+7

for all real numbers $t$ . What is the value of $3\bigl(k-m\bigr)$ ?

An identity is an equation that holds for every input. When a function has an expression inside its parentheses, substitute that whole expression into its rule. Then use a Desmos regression at several inputs to find the unknown constants. One input cannot determine two constants, and finding the constants is not the same as evaluating the expression the question asks for.

Hints

Hint 1
The expression inside $g(3t-5)$ is the whole input to $g$ . In the rule $g(t)=kt+m$ , what replaces the input being multiplied by $k$ ?
Hint 2
The equation is an identity: it holds for every $t$ . One input gives only one equation, but there are two unknown constants. How could you use several inputs to determine both?
Hint 3
A Desmos regression uses $\sim$ to fit the constants across several inputs. Once it reports $k$ and $m$ , type the entire requested expression $3(k-m)$ rather than stopping at $k-m$ .

Step-by-step

Approach 1: Fit the constants in Desmos
Step 1
Substitute the whole input
The rule $g(t)=kt+m$ says to multiply an input by $k$ , then add $m$ . In $g(3t-5)$ , the whole input is $3t-5$ , so the given equation becomes $k(3t-5)+m=2t+7$ . Keep the $-5$ inside the parentheses: it gets multiplied by $k$ too.
Step 2
Fit both constants at several inputs
Because the equation holds for all real $t$ , let $t_1$ be the list $1,2,3,4,5$ . Type the two lines shown. The $\sim$ tells Desmos to fit $k$ and $m$ so the sides agree across the list. Under PARAMETERS, Desmos shows $k\approx0.6667$ and $m\approx10.3333$ . One input cannot determine two unknown constants; several inputs can.
Step 3
Evaluate what the question asks
Keep those lines and type the full expression $3(k-m)$ . Desmos uses the fitted constants and displays $-29$ . So $3(k-m)=-29$ ; don't multiply by $3$ again. Choice B.

Approach 2: Match the terms by hand
Step 1
Expose the terms involving $t$
Instead of fitting inputs, use the fact that both sides agree for every $t$ . Distribute $k$ in $k(3t-5)+m=2t+7$ to separate the term involving $t$ from the terms without it: $3kt-5k+m=2t+7$ .
Step 2
Match the coefficients of $t$
A coefficient is the number multiplying a variable. The coefficients of $t$ must match because the equation holds for every $t$ : $3k=2$ . Divide by $3$ : $k=\frac{2}{3}$ .
Step 3
Match the constant terms
A constant term has no $t$ . Those terms must match too: $-5k+m=7$ . Replace $k$ with $\frac{2}{3}$ : $-5\left(\frac{2}{3}\right)+m=7$ . Multiply: $-\frac{10}{3}+m=7$ . Add $\frac{10}{3}$ to both sides: $m=7+\frac{10}{3}$ . Write $7$ in thirds: $m=\frac{21}{3}+\frac{10}{3}$ . Add: $m=\frac{31}{3}$ .
Step 4
Combine the constants in the requested order
Now type $3\left(\frac{2}{3}-\frac{31}{3}\right)$ into Desmos. It displays $-29$ , the value of $3(k-m)$ . The order of subtraction matters. Choice B.

Question 312·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this