For a constant k, the graph of the linear equation kx - 4y = 15 in the xy-plane contains t… — SAT

For a constant $k$ , the graph of the linear equation $kx - 4y = 15$ in the $xy$ -plane contains the point $(3, 3)$ . What is the slope of the line?

Answer: 2.25

Answer

2.25

Substituting the coordinates of the point

(3, 3)

into the given equation yields

k(3) - 4(3) = 15

, which simplifies to

3k - 12 = 15

. Adding

12

to both sides gives

3k = 27

, and dividing by

3

gives

k = 9

. Substituting

k = 9

back into the original equation gives

9x - 4y = 15

. To find the slope of the line, the equation can be written in slope-intercept form,

y = mx + b

, by isolating

y

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing all terms by

-4

yields

y = \frac{9}{4}x - \frac{15}{4}

, or

y = 2.25x - 3.75

. Therefore, the slope of the line is

\frac{9}{4}

, or

2.25

Step-by-Step Solution

Substitute the point

(3, 3)

into the equation

kx - 4y = 15

3k - 12 = 15

Since the graph of the equation contains the point

(3, 3)

, substituting

x = 3

and

y = 3

must satisfy the equation.

Solve for the constant

k

k = 9

Adding

12

to both sides of the equation gives

3k = 27

. Dividing both sides by

3

yields

k = 9

Substitute

k = 9

into the equation and rewrite it in slope-intercept form (

y = mx + b

)

y = \frac{9}{4}x - \frac{15}{4}

Substituting

k = 9

gives

9x - 4y = 15

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing both sides by

-4

isolates

y

Identify the slope from the slope-intercept form

Slope =

2.25

(or

\frac{9}{4}

)

In the slope-intercept form

y = mx + b

, the coefficient

m

represents the slope of the line.

Key Concept

Determining the slope of a line from a linear equation by first finding an unknown constant using a given point on the line.

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