In the equation below, is a constant. For what value of does the equation have no solution?

Solution available with step-by-step explanation

SAT Linear Equations in One Variable Question 132 | Free SAT Prep | Aniko

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Question 132·Hard·Linear Equations in One Variable

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In the equation below, $p$ is a constant.

$(p + 3)x + 2 = (2p - 5)x - 4$

For what value of $p$ does the equation have no solution?

For SAT questions asking when an equation has no solution or infinitely many solutions, rewrite the equation so you can clearly see the coefficient of $x$ and the constant term on each side. For a linear equation $ax+b=cx+d$ , remember: if $a=c$ but $b\ne d$ , there is no solution; if $a=c$ and $b=d$ , there are infinitely many solutions; otherwise, there is exactly one solution. Apply this pattern directly by equating the $x$ -coefficients, solve for the parameter, and quickly check that the constants are unequal for a no-solution case.

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Think about the type of equation

This equation is linear in $x$ . Imagine each side as a line: when would two lines never intersect?

Compare the structure of both sides

Look at the coefficient of $x$ and the constant term on each side. How must these compare for two lines to be parallel but not the same line?

Write an equation for the coefficients of $x$

Set the coefficient of $x$ on the left equal to the coefficient of $x$ on the right and solve that equation for $p$ .

Check the constants

After you find $p$ from the coefficient equation, make sure the constants (the terms without $x$ ) are still different so there is truly no solution.

Desmos Guide

Enter both sides as functions of x

In one line, type (p+3)x+2 and accept the slider for p. In the next line, type (2p-5)x-4, using the same p slider. You will see two lines whose positions depend on p.

Use the slider to find when the lines are parallel but distinct

Move the p slider and watch the graphs. Look for the value of p where the two lines have exactly the same slope (they never get closer or farther apart) but have different $y$ -intercepts (they cross the $y$ -axis at different points). The corresponding p value is the one that makes the original equation have no solution.

Step-by-step Explanation

Recall when a linear equation has no solution

A linear equation like $ax+b=cx+d$ has no solution when the two sides describe parallel but distinct lines. In algebra terms, that happens when the coefficients of $x$ are equal ( $a=c$ ), but the constant terms are different ( $b\ne d$ ).

Identify coefficients and constants in the given equation

Rewrite the equation highlighting the $x$ -terms and constants:

(p+3)x + 2 = (2p - 5)x - 4

The coefficient of $x$ on the left is $p+3$ .
The coefficient of $x$ on the right is $2p-5$ .
The constant on the left is $2$ .
The constant on the right is $-4$ (which is clearly different from $2$ ).

Set the $x$ -coefficients equal for parallel lines

For the equation to have no solution, the $x$ -coefficients must match (so the lines are parallel), while the constants stay different (which they already are: $2\ne -4$ ).

Set the coefficients of $x$ equal:

p + 3 = 2p - 5

Solve for the value of $p$

Now solve the equation from the previous step:

\begin{aligned} p + 3 &= 2p - 5 \\ 3 + 5 &= 2p - p \\ 8 &= p \end{aligned}

So the equation has no solution when $p=8$ , which corresponds to answer choice C.

Strategy

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation