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 point $(2, 7)$ . If this graph is translated $3$ units to the right and $4$ units down, the resulting graph passes through the point $(4, 2)$ . What is the value of $b$ ?

Answer: 5

Answer

The correct answer is

5

. Since the original graph passes through

(2, 7)

and undergoes a translation of

3

units to the right and

4

units down, the point

(2, 7)

is translated to

(2 + 3, 7 - 4) = (5, 3)

on the new graph. The translated graph also passes through

(4, 2)

. Using the slope formula on the two points

(5, 3)

and

(4, 2)

of the translated line gives a slope of

1

. Since a translation preserves the slope, the original line also has a slope of

1

. Substituting

m = 1

and

(2, 7)

into the original equation

y = mx + b

yields

7 = 1(2) + b

, which simplifies to

b = 5

Step-by-Step Solution

Determine a point on the translated graph by translating the given point

(2, 7)

on the original graph.

The point

(5, 3)

is on the translated graph.

Since the entire graph is translated

3

units to the right and

4

units down, every point on the original graph is translated by the same vector

(+3, -4)

Calculate the slope of the translated graph using the points

(5, 3)

and

(4, 2)

The slope

m = 1

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the change in y divided by the change in x.

Find the slope of the original graph.

The slope of the original graph is also

1

Translating a line horizontally and vertically shifts its position but does not alter its slope.

Substitute the slope

m = 1

and the point

(2, 7)

into the slope-intercept form of the original line,

y = mx + b

, to solve for

b

b = 5

This determines the value of the y-intercept of the original line.

Key Concept

Linear Equations in Two Variables

Question

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