SAT Linear Equations in One Variable Question 114 | Free SAT Prep | Aniko

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Question 114·Hard·Linear Equations in One Variable

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

For SAT questions about linear equations with parameters and "infinitely many solutions" or "no solution," first rewrite both sides in the standard form $mx + b$ . Then use the idea that two linear expressions are identical for all $x$ only if their $x$ -coefficients and constants match exactly. Set the coefficients equal to each other and the constants equal to each other, solve for the parameters, and then match these results to the answer choices. This approach is much faster and more reliable than plugging in random values of $x$ .

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Think about what "infinitely many solutions" means

For a linear equation in one variable to be true for every value of $x$ , what must be true about the two expressions on each side of the equals sign?

Put both sides in the same form

Distribute $s$ on the right-hand side of $s(x - 4)$ so that both sides look like $(\text{something})x + (\text{something})$ and then compare those parts.

Match like parts

Once both sides are written as $(\text{coefficient})x + (\text{constant})$ , set the coefficients of $x$ equal to each other and the constants equal to each other. Then see what that tells you about $s$ and $t$ .

Desmos Guide

Graph both sides with sliders for s and t

In Desmos, enter y = (5/3)x + t and y = s(x - 4). When prompted, create sliders for s and t so you can adjust their values.

Make the lines coincide

Adjust the slider for s until the two lines have the same slope (they look parallel). Then adjust the slider for t so that the two lines lie exactly on top of each other. Note the values of s and t at this point; these are the values where the equation has infinitely many solutions.

Check the relationship between s and t

In Desmos, type an expression like t + 4s or t / s using the values from the sliders to see how $t$ is related to $s$ . Compare this relationship with statements I, II, and III to confirm which must be true.

Step-by-step Explanation

Use the condition "infinitely many solutions"

If an equation of the form $ax + b = cx + d$ has infinitely many solutions, then both sides must represent the same linear expression. That means:

The coefficients of $x$ must be equal.
The constant terms (the numbers without $x$ ) must be equal.

We will apply this idea to the given equation.

Rewrite the equation so both sides look like $mx + b$

Start with the given equation:

\frac{5}{3}x + t = s(x - 4)

Distribute $s$ on the right-hand side:

\frac{5}{3}x + t = sx - 4s

Now both sides are linear expressions in the form (coefficient of $x$ ) times $x$ plus a constant.

Match coefficients and constants

Since the equation has infinitely many solutions, the expressions on both sides must be identical for all $x$ :

The coefficients of $x$ must match: the left side has coefficient $5/3$ , and the right side has coefficient $s$ . So

s = \frac{5}{3}

The constant terms must match: the left side has constant $t$ , and the right side has constant $-4s$ . So

t = -4s

From this, you can already decide which of statements I and II must be true.

Find $t$ and determine which statements must be true

We already have:

$s = 5/3$
$t = -4s$

Substitute $s = 5/3$ into $t = -4s$ :

t = -4\left(\frac{5}{3}\right) = -\frac{20}{3}

Now compare with the statements:

I. $s = 5/3$ — this matches what we found.
II. $t = -4s$ — this also matches what we found.
III. $t = -20/3$ — this matches the value of $t$ we just calculated.

So all three statements (I, II, and III) must be true, which corresponds to answer choice D: I, II, and III.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation