In the xy-plane, a line with a positive slope m passes through the points (2, 5) and (m, 1… — SAT

In the $xy$ -plane, a line with a positive slope $m$ passes through the points $(2, 5)$ and $(m, 13)$ . Which of the following is the $y$ -intercept of this line?

A
$(0, 3)$
B
$(0, 9)$
$(0, -3)$ Answer
D
$(0, 1)$

Answer

(0, -3)

The correct answer is the point

(0, -3)

. The slope of the line passing through

(2, 5)

and

(m, 13)

is expressed as

m = \frac{13 - 5}{m - 2} = \frac{8}{m - 2}

. Multiplying by

m - 2

yields the quadratic equation

m^2 - 2m - 8 = 0

. Factoring this equation gives

(m - 4)(m + 2) = 0

. Since the slope

m

is given as positive, we select

m = 4

. Using the point-slope form with

m = 4

and the point

(2, 5)

yields the equation

y - 5 = 4(x - 2)

, which simplifies to

y = 4x - 3

. The

y

-intercept of this line is found by setting

x = 0

, giving

y = -3

, or the coordinate point

(0, -3)

Step-by-Step Solution

Use the slope formula to express the slope of the line passing through

(2, 5)

and

(m, 13)

in terms of

m

m = \frac{13 - 5}{m - 2} \implies m = \frac{8}{m - 2}

The slope of a line is defined as the ratio of the change in

y

to the change in

x

between any two points on the line.

Solve the equation

m = \frac{8}{m - 2}

for

m

by converting it into a quadratic equation.

m(m - 2) = 8 \implies m^2 - 2m - 8 = 0 \implies (m - 4)(m + 2) = 0

, which gives

m = 4

m = -2

. Since the slope is positive,

m = 4

Multiplying both sides by the denominator clears the fraction and allows us to solve for the unknown parameter

m

using factoring.

Determine the equation of the line using the point-slope form with

m = 4

and the point

(2, 5)

y - 5 = 4(x - 2) \implies y = 4x - 8 + 5 \implies y = 4x - 3

The point-slope form provides a direct way to write the linear equation using a known point and the calculated slope.

Find the

y

-intercept by evaluating the line equation at

x = 0

y = 4(0) - 3 \implies y = -3

, which corresponds to the point

(0, -3)

The

y

-intercept of any line is the point where the line crosses the

y

-axis, which occurs at

x = 0

Key Concept

Linear Equations in Two Variables

Alternative Method

Alternatively, once the slope

m = 4

is found, we can write the equation of the line in slope-intercept form

y = 4x + b

. Substituting the coordinates of the point

(2, 5)

into the equation gives

5 = 4(2) + b \implies 5 = 8 + b

. Solving for

b

gives

b = -3

. Since

b

is the

y

-intercept, the point is

(0, -3)

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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