In the -plane, the line given by intersects the coordinate axes at two points. If is a positive integer and the difference between the -intercept and the -intercept of the line is , what is the value of ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 33 | Free SAT Prep | Aniko

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Question 33·Hard·Linear Equations in Two Variables

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In the $xy$ -plane, the line given by $5x + ky = 20$ intersects the coordinate axes at two points. If $k$ is a positive integer and the difference between the $x$ -intercept and the $y$ -intercept of the line is $3$ , what is the value of $k$ ?

Your answer

(Express the answer as an integer)

For line-intercept problems, immediately find the x-intercept by setting $y=0$ and the y-intercept by setting $x=0$ . Once you have both intercept values (even if one includes a parameter like $k$ ), translate any word condition such as "difference is 3" into a simple equation, often using an absolute value to represent a positive difference. Solve this equation for the parameter, then apply any extra conditions (like "positive integer") to select the valid solution. This direct algebraic approach is much faster and less error-prone than trying to reason purely in words or by drawing a rough graph.

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Start with intercepts

How do you usually find the x-intercept and y-intercept of a line given by an equation like $5x + ky = 20$ ?

Express the intercepts as numbers

When you set $y=0$ , what equation do you get for $x$ ? When you set $x=0$ , what equation do you get for $y$ ? Write the intercepts as $4$ and $20/k$ .

Turn the word "difference" into an equation

Use the numbers $4$ and $20/k$ and write an equation that says their (positive) difference is 3. Remember that an absolute value like $|a-b|$ represents the distance between $a$ and $b$ on the number line.

Check the condition on k

After solving the equation, you may get more than one value for $k$ . Which of them is a positive integer, as required by the problem?

Desmos Guide

Graph the difference between intercepts as a function of k

In Desmos, use $x$ in place of $k$ . Enter the function

$y = \left|4 - \dfrac{20}{x}\right|$

This represents the (positive) difference between the x-intercept (4) and the y-intercept ( $20/x$ ) as $x$ varies.

Compare this function to the given difference

In a new line, enter

$y = 3$

This is the horizontal line representing the required difference of 3.

Find the value of k from the graph

Look for the intersection points of the two graphs. The x-coordinate(s) of these intersection points are the possible values of $k$ . Among those, choose the one that is a positive integer; that is the value of $k$ that solves the problem.

Step-by-step Explanation

Find the x- and y-intercepts in terms of k

To find the intercepts, remember:

For the x-intercept, set $y=0$ .
For the y-intercept, set $x=0$ .

Apply this to $5x + ky = 20$ :

x-intercept: set $y=0$ :

$5x = 20$ , so $x = 4$ .

So the x-intercept is 4.
y-intercept: set $x=0$ :

$ky = 20$ , so $y = \dfrac{20}{k}$ .

So the y-intercept is $\dfrac{20}{k}$ .

Translate the "difference is 3" condition into an equation

The two intercept values are the numbers $4$ and $\dfrac{20}{k}$ .

"The difference between" two numbers usually refers to the positive difference (their distance on the number line), so we use an absolute value:

\left|4 - \frac{20}{k}\right| = 3

This equation says the distance between $4$ and $\dfrac{20}{k}$ is 3.

Solve the absolute value equation for k

To solve

\left|4 - \frac{20}{k}\right| = 3,

consider the two possible cases:

Case 1: $4 - \dfrac{20}{k} = 3$

Then

4 - 3 = \frac{20}{k} \\[0.7em] 1 = \frac{20}{k}

Multiply both sides by $k$ :

k = 20

Case 2: $\dfrac{20}{k} - 4 = 3$

Then

\frac{20}{k} = 7 \\[0.7em] 20 = 7k \\[0.7em] k = \frac{20}{7}

So the two possible values from the algebra are $k = 20$ and $k = \dfrac{20}{7}$ .

Use the condition on k to choose the valid value

The problem states that $k$ is a positive integer.

$k = \dfrac{20}{7}$ is not an integer.
$k = 20$ is a positive integer.

Therefore, the value of $k$ that satisfies all conditions is $20$ .

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation