SAT Linear Equations in One Variable Question 17 | Free SAT Prep | Aniko

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Question 17·Medium·Linear Equations in One Variable

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$5\bigl(x - 3\bigr) = 5\bigl(x + 2\bigr)$

How many solutions does the given equation have?

For “how many solutions” questions with linear equations, first simplify both sides (distribute, combine like terms, or divide common factors), then bring all variable terms to one side and constants to the other. If you get a specific $x$ value, there is exactly one solution; if the variables cancel and you get a true statement like $0 = 0$ , there are infinitely many solutions; if the variables cancel and you get a false statement like $-3 = 2$ , there are zero solutions. This approach avoids extra computation and quickly classifies the equation.

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Clear the common factor

Look at $5(x - 3) = 5(x + 2)$ . What simple operation can you do to both sides to get rid of the 5?

Compare the simplified sides

After you remove the 5, you should get an equation with $x$ on both sides. What happens if you subtract $x$ from both sides?

Check if the final statement can be true

When the variable disappears, you are left with a statement involving only numbers. Ask yourself: can that numerical statement ever be true? What does that tell you about the number of solutions?

Desmos Guide

Enter each side as a separate function

In Desmos, type y = 5(x - 3) on one line and y = 5(x + 2) on another line to graph both sides of the equation as separate lines.

Use the graph to count solutions

Look for intersection points of the two lines. Each intersection corresponds to a solution of the equation. The number of intersection points tells you how many solutions the equation has.

Step-by-step Explanation

Simplify the equation

Start with the given equation:

5(x - 3) = 5(x + 2)

Since both sides are multiplied by 5 (and 5 is not 0), divide both sides by 5 to simplify:

x - 3 = x + 2

Eliminate the variable term

Now remove $x$ from both sides by subtracting $x$ from each side of the equation:

x - 3 - x = x + 2 - x

This simplifies to:

-3 = 2

Interpret the result and answer the question

The equation $-3 = 2$ is never true, no matter what $x$ is. That means there is no value of $x$ that makes the original equation true.

So, the equation has zero solutions, and the correct answer is D) Zero.

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Step-by-step Explanation

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Step-by-step Explanation