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 $(2, 15)$ and $(6, 7)$ . What is the value of $b$ ?

Answer: 19

Answer

The value of

b

19

The slope of the line is found using the two given points:

m = \frac{7 - 15}{6 - 2} = -2

. Substituting the slope

m = -2

and the point

(2, 15)

into the equation

y = mx + b

gives

15 = -2(2) + b

, which simplifies to

15 = -4 + b

. Adding

4

to both sides yields

b = 19

Step-by-Step Solution

Calculate the slope of the line passing through

(2, 15)

and

(6, 7)

m = \frac{7 - 15}{6 - 2} = \frac{-8}{4} = -2

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

Substitute the slope

m = -2

and the coordinates of one point, such as

(2, 15)

, into the equation

y = mx + b

to find

b

15 = -2(2) + b \implies 15 = -4 + b

Since the point lies on the line, its coordinates must satisfy the equation of the line.

Solve the equation for

b

b = 19

Add

4

to both sides of the equation to isolate the variable

b

Key Concept

Determining the equation of a line given two points.

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