In the standard (x, y) coordinate plane, a line passes through the points (2, 5) and (6, k… — ACT

In the standard $(x, y)$ coordinate plane, a line passes through the points $(2, 5)$ and $(6, k)$ . If the $y$ -intercept of this line is $-1$ , what is the value of $k$ ?

Answer: 17

Answer

The value of

k

is 17.

The correct answer is 17. The y-intercept of

-1

gives the point

(0, -1)

. The slope between

(0, -1)

and

(2, 5)

\frac{5 - (-1)}{2 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting the point

(6, k)

into the equation gives

k = 3(6) - 1 = 17

Step-by-Step Solution

Identify the coordinates of the y-intercept.

The point is

(0, -1)

The y-intercept is the point where the line crosses the y-axis, which occurs when the x-coordinate is 0.

Calculate the slope (

m

) of the line using the points

(0, -1)

and

(2, 5)

m = 3

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

, we find

m = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3

Write the equation of the line.

y = 3x - 1

Using the slope-intercept form

y = mx + b

, where the slope

m = 3

and the y-intercept

b = -1

Substitute the point

(6, k)

into the line equation to solve for

k

k = 17

Since the point

(6, k)

lies on the line, substituting

x = 6

and

y = k

yields

k = 3(6) - 1 = 18 - 1 = 17

Key Concept

Linear Equations and Graphing

Alternative Method

Since the slope of a straight line is constant, the slope between

(0, -1)

and

(2, 5)

must equal the slope between

(2, 5)

and

(6, k)

. Setting the slopes equal:

\frac{5 - (-1)}{2 - 0} = \frac{k - 5}{6 - 2} \implies 3 = \frac{k - 5}{4} \implies 12 = k - 5 \implies k = 17

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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