In the xy-plane, the graph of the linear equation y = mx + b, where m and b are constants,… — SAT

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the points $(-2, 5)$ and $(4, 7)$ . What is the value of $m + b$ ?

$6$ Answer
B
$-2$
C
$4$
D
$\frac{14}{3}$

Answer

The value of

m + b

6

The correct answer is

6

. By finding the slope

m

using the change in

y

divided by the change in

x

, we get

m = \frac{1}{3}

. Using the slope-intercept form with the point

(4, 7)

allows us to solve for

b = \frac{17}{3}

. Adding the two values together yields

m + b = \frac{1}{3} + \frac{17}{3} = 6

Step-by-Step Solution

Calculate the slope

m

of the line using the two points

(-2, 5)

and

(4, 7)

m = \frac{7 - 5}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

Substitute the slope

m = \frac{1}{3}

and the coordinates of one point, such as

(4, 7)

, into the slope-intercept equation

y = mx + b

to solve for

b

7 = \frac{1}{3}(4) + b \Rightarrow 7 = \frac{4}{3} + b \Rightarrow b = \frac{17}{3}

Substituting a known point and the slope into the slope-intercept form allows us to isolate and solve for the constant

y

-intercept.

Add the calculated values of

m

and

b

to find

m + b

m + b = \frac{1}{3} + \frac{17}{3} = \frac{18}{3} = 6

The question asks for the sum of the slope

m

and the

y

-intercept

b

Key Concept

Linear Equations in Two Variables

Question

Answer

Step-by-Step Solution

Key Concept

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