Consider the following system of linear equations. If the graphs of the two equations are perpendicular in the -plane, what is the value of ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 111 | Free SAT Prep | Aniko

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Question 111·Hard·Linear Equations in Two Variables

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Consider the following system of linear equations.

\begin{aligned} 5x + k y &= 10 \\ 2x - 5y &= 7 \end{aligned}

If the graphs of the two equations are perpendicular in the $xy$ -plane, what is the value of $k$ ?

Your answer

(Express the answer as an integer)

For SAT questions about perpendicular lines given in standard form, quickly convert each equation to slope-intercept form so you can read off the slopes. Remember that for non-vertical lines, “perpendicular” means the slopes are negative reciprocals, which is equivalent to saying their product is $-1$ . Set up a simple equation using $m_1 m_2 = -1$ , solve carefully for the unknown parameter (watching signs closely), and avoid extra work by simplifying fractions step by step.

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Focus on slopes

Perpendicularity in the coordinate plane is about slopes. Try to rewrite each equation in the form $y = mx + b$ so you can see the slope $m$ .

Isolate y in the first equation

From $5x + ky = 10$ , solve for $y$ in terms of $x$ and $k$ . The coefficient of $x$ in that expression is the slope of the first line.

Isolate y in the second equation

From $2x - 5y = 7$ , solve for $y$ to identify the slope of the second line.

Use the perpendicular slopes relationship

Once you have both slopes, remember that for non-vertical lines that are perpendicular, their slopes multiply to $-1$ . Set up that equation and solve for $k$ .

Desmos Guide

Express slopes in terms of k

In Desmos, type y = (-5/x)*(2/5) and y = -1. The first expression is the product of the two slopes.

Find the intersection

Click on the intersection point. The x-coordinate gives the value of $k$ that makes the lines perpendicular (product of slopes = $-1$ ).

Step-by-step Explanation

Recall the condition for perpendicular lines

For two non-vertical lines with slopes $m_1$ and $m_2$ , the lines are perpendicular if and only if

m_1 \times m_2 = -1.

So our goal is to find the slopes of the two given lines and then set their product equal to $-1$ .

Find the slope of the first line

Start with the first equation:

5x + ky = 10.

Solve for $y$ :

ky = -5x + 10 \\[0.7em] y = \frac{-5}{k}x + \frac{10}{k}.

So the slope of the first line is

m_1 = \frac{-5}{k}.

Find the slope of the second line

Start with the second equation:

2x - 5y = 7.

Solve for $y$ :

-5y = -2x + 7 \\[0.7em] y = \frac{2}{5}x - \frac{7}{5}.

So the slope of the second line is

m_2 = \frac{2}{5}.

Use the perpendicular condition to form an equation in k

Use the condition $m_1 \times m_2 = -1$ with $m_1 = -5/k$ and $m_2 = 2/5$ :

\frac{-5}{k} \cdot \frac{2}{5} = -1.

Multiply the fractions on the left:

\frac{-10}{5k} = -1 \Rightarrow \frac{-2}{k} = -1.

Solve for k

From

\frac{-2}{k} = -1,

we can multiply both sides by $k$ :

-2 = -k,

k = 2.

Therefore, the value of $k$ is $2$ .

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