Question 566·Hard·Algebra

0 / 700

The linear equation $\dfrac{5}{3}x + t = s\,(x - 4)$ , where $s$ and $t$ are constants, has infinitely many solutions.

Which of the following must be true?

I. $s = \dfrac{5}{3}$

II. $t = -4s$

III. $t = -\dfrac{20}{3}$

The cue infinitely many solutions in a linear equation means both sides must be identical for every value of $x$ . Distribute first, then match the numbers multiplying $x$ and the terms without $x$ . Matching only the $x$ -terms can leave a false statement, which gives no solutions instead.

Hints

Hint 1
An identity is an equation true for every value of $x$ . For infinitely many solutions, the two sides must become the same expression. What do you get when you multiply $s$ by both terms in $(x-4)$ ?
Hint 2
The coefficient of $x$ is the number multiplying it. If the two sides agree for every $x$ , their coefficients of $x$ must match. What coefficient do you see on each side after distributing?
Hint 3
Next, compare the constant terms, the parts without $x$ . They must match too; otherwise the $x$ -terms cancel and leave a false equation. What does their match tell you about $t$ ?

Step-by-step

Match both parts of the equation
Step 1
Distribute on the right
Multiply $s$ by both terms in $(x-4)$ :
$\frac{5}{3}x+t=sx-4s.$
Step 2
Match the coefficients of $x$
Infinitely many solutions require the $x$ -coefficients and the constant terms to match. The coefficient is the number multiplying $x$ . Here those numbers are $\frac{5}{3}$ and $s$ , so matching the $x$ -terms gives
$s=\frac{5}{3}.$
Step 3
Match the constant terms
The constant terms are the parts without $x$ : $t$ on the left and $-4s$ on the right. They must match too, so $t=-4s$ . Matching only the $x$ -terms could leave a false equation after those terms cancel.
Step 4
Check the constants in Desmos
Since every value of $x$ must work, check several at once. Type $x_1=[0...4]$ , then type $\frac{5}{3}x_1+t\sim s(x_1-4)$ . The symbol $\sim$ tells Desmos to fit $s$ and $t$ to those inputs. Under PARAMETERS, it shows $s\approx1.6667$ and $t\approx-6.6667$ , consistent with both matches.
Step 5
Get the exact value of $t$
Type $t$ on a new Desmos line and use its fraction button. It shows $t=-\frac{20}{3}$ , the exact value of $-4\left(\frac{5}{3}\right)$ . So the third statement follows from the first two, and all three must be true. Choice D.

Question 566·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this