A line in the standard (x,y) coordinate plane is defined by the equation kx + 5y = 20, whe… — ACT

A line in the standard $(x,y)$ coordinate plane is defined by the equation $kx + 5y = 20$ , where $k$ is a constant. If the $x$ -intercept of this line is $6$ units greater than its $y$ -intercept, what is the value of $k$ ?

A
$\frac{1}{2}$
$2$ Answer
C
$\frac{10}{3}$
D
$10$
E
$-10$

Answer

2

To find the

y

-intercept of the line

kx + 5y = 20

, set

x = 0

, which yields

5y = 20

, or

y = 4

. The problem states that the

x

-intercept is

6

units greater than the

y

-intercept, so the

x

-intercept is

4 + 6 = 10

. Substituting the point

(10, 0)

back into the original equation gives

k(10) + 5(0) = 20

. Solving for

k

gives

10k = 20

, which simplifies to

k = 2

Step-by-Step Solution

Find the

y

-intercept of the line.

The

y

-coordinate of the

y

-intercept is

4

To find the

y

-intercept, set

x = 0

in the equation

kx + 5y = 20

, which gives

5y = 20

, so

y = 4

Determine the

x

-intercept of the line based on the given relationship.

The

x

-coordinate of the

x

-intercept is

10

The problem states that the

x

-intercept is

6

units greater than the

y

-intercept. Since the

y

-intercept value is

4

, the

x

-intercept value is

4 + 6 = 10

Substitute the

x

-intercept coordinates into the equation to solve for

k

k = 2

The

x

-intercept is the point

(10, 0)

. Substituting

x = 10

and

y = 0

into the equation

kx + 5y = 20

gives

k(10) + 5(0) = 20

, which simplifies to

10k = 20

. Dividing both sides by

10

yields

k = 2

Key Concept

Finding intercepts of a linear equation in standard form and using coordinate substitution to solve for an unknown coefficient.

Question

Answer

Step-by-Step Solution

Key Concept

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