In the standard (x, y) coordinate plane, line L_1 is perpendicular to the line represented… — ACT

In the standard $(x, y)$ coordinate plane, line $L_1$ is perpendicular to the line represented by the equation $3x - 2y = 6$ . Line $L_2$ is parallel to $L_1$ and passes through the midpoint of the line segment with endpoints $(-1, 5)$ and $(3, -1)$ . If the equation of $L_2$ is written in the form $Ax + By = 8$ , where $A$ and $B$ are integers, what is the value of $A + B$ ?

Cevap: 5

Cevap

The value of

A + B

is 5.

The given line

3x - 2y = 6

has a slope of

\frac{3}{2}

. A perpendicular line

L_1

has a slope of

-\frac{2}{3}

. A line

L_2

parallel to

L_1

also has a slope of

-\frac{2}{3}

. The midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

is calculated as

\left(\frac{-1+3}{2}, \frac{5-1}{2}\right) = (1, 2)

. Using the point-slope form with the slope

-\frac{2}{3}

and point

(1, 2)

yields

y - 2 = -\frac{2}{3}(x - 1)

, which simplifies to

2x + 3y = 8

. Thus,

A = 2

B = 3

, and

A + B = 5

Adım Adım Çözüm

Find the slope of the given line

3x - 2y = 6

Slope is

\frac{3}{2}

Convert the equation to slope-intercept form

y = mx + b

to find the slope.

Determine the slope of line

L_1

Slope is

-\frac{2}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

Determine the slope of line

L_2

Slope is

-\frac{2}{3}

Parallel lines have the same slope.

Calculate the midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

Midpoint is

(1, 2)

Use the midpoint formula:

\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Write the equation of line

L_2

and convert it to standard form.

Equation is

2x + 3y = 8

Use the point-slope form with slope

-\frac{2}{3}

and midpoint

(1, 2)

, then rearrange to match the target constant 8 on the right-hand side.

Identify

A

and

B

, and compute their sum.

A = 2

B = 3

, and

A + B = 5

Match the coefficients of

2x + 3y = 8

Ax + By = 8

and add them.

Anahtar Kavram

Determining equations of parallel and perpendicular lines on the coordinate plane using slopes and midpoints.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru