Question 143·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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s = \frac{5t - 1}{2t + 3}

The equation above relates the distinct real numbers $s$ and $t$ with $2t + 3 \neq 0$ . Which equation correctly expresses $t$ in terms of $s$ ?

In a rational equation, the letter you need may appear above and below a fraction bar. On a timed test, choose an allowed value for the other letter, use a Desmos regression to find the target, and compare the choices. A test can eliminate wrong formulas, but a match alone doesn't prove a formula always works. For a general solution, clear the denominator, then factor out the target letter.

Hints

Hint 1
To test formulas with letters, give $s$ a fixed value, such as $1$ . The correct formula must give the same $t$ as the original equation at that value. Check afterward that $s$ and $t$ are distinct.
Hint 2
A regression asks Desmos to make two sides equal. Replace $t$ with $x_1$ and $=$ with $\sim$ so Desmos solves for $x_1$ . What value does it find when $s=1$ ?
Hint 3
Compare each choice's output with the same $t$ from the original equation. If two choices match at one value of $s$ , test another allowed value; one matching test doesn't prove they always agree.

Step-by-step

Approach 1: Test the formulas in Desmos
Step 1
Find $t$ for an allowed value of $s$
Pick $s=1$ to keep the choices short. Type $s=1$ , then $\frac{5x_1-1}{2x_1+3}\sim s$ . Here $x_1$ stands for $t$ , and $\sim$ tells Desmos to solve the original equation. Under PARAMETERS, Desmos shows $x_1\approx1.333$ . That's different from $s=1$ , so the pair meets the distinct-numbers condition; the fraction is defined there too.
Step 2
Compare the choices at $s=1$
Type the right side of each choice on a new line, in the order listed. Desmos prints $-3$ , about $-0.286$ , about $-1.333$ , and about $1.333$ . Only the last formula gives the $t$ required by the original equation at this allowed value; the other three cannot be correct for every allowed pair. The equation expressing $t$ in terms of $s$ is $t=\frac{1+3s}{5-2s}$ . Choice D.

Approach 2: Derive the formula for every allowed pair
Step 1
Clear the denominator
The given denominator is not zero, so multiply both sides by the whole expression $2t+3$ :
$s(2t+3)=5t-1.$
Step 2
Distribute $s$
Multiply $s$ by each term inside the parentheses:
$2st+3s=5t-1.$
Step 3
Gather the $t$ -terms
Treat $s$ as fixed while you get $t$ alone. Add $1$ to both sides:
$2st+3s+1=5t.$
Subtract $2st$ from both sides so both $t$ -terms are on the right:
$3s+1=5t-2st.$
Step 4
Factor out $t$
Use factoring to write the two right-side terms as one multiple of $t$ :
$3s+1=t(5-2s).$
The minus sign matters: it comes from the $-2st$ you moved to the right.
Step 5
Divide to isolate $t$
You can divide by $5-2s$ : if it were zero, $s$ would be $\frac52$ , making the left side $3s+1$ positive while the right side $t(5-2s)$ is zero. Divide by the entire factor $5-2s$ :
$t=\frac{3s+1}{5-2s}.$
This gives $t$ in terms of $s$ for every pair allowed by the original equation. Choice D.

Question 143·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this