Line ℓ passes through (-2, 5) and (4, -7). Write an equation for the line that passes through (1, 3) and is parallel to line ℓ. Leave in slope-intercept form.

Answer 1

Problem

Line L passes through (-2 , 5) and (4, -7). Write an equation for the line that passes through (1, 3) and is parallel to line L. Express your answer in slope-intercept form.

Introduction

Recall that slope-intercept form has the following appearance:

y = mx + b

... where m is the slope of the line and b is the y-intercept (where x = 0).

Similarly, point-slope form looks like:

y - y₁ = m(x - x₁)

Additionally, recall that the slope of a line, m, is equal to:

m = (y₂ - y₁) / (x₂ - x₁)

Step 1

Since we are given two points on line L, (-2, 5) and (4, -7), we can use the slope equation to find our m:

m = (-7 - 5) / (4 - -2)

m = -2

Step 2

Using the slope we found from part 1, we can now use the point-slope formula to find the equation of the parallel line that passes through (1, 3).

Remember: Since this line must be parallel to line L, m = -2 for both lines.

y - y₁ = m(x - x₁)

y - 3 = -2(x - 1)

Step 3

Simplify the point-slope equation from step 2 into the slope-intercept form y = mx + b:

y = 3 - 2(x - 1)

y = 3 - 2x + 2

Therefore, y = -2x + 5.

Solution

y = -2x + 5.

Answer 2

To write the equation of a line that passes through the point (1,3)(1, 3) and is parallel to the line ℓ\ell passing through (−2,5)(-2, 5) and (4,−7)(4, -7), we will follow these steps:

Step 1: Find the slope of line ℓ\ell

The slope of a line is calculated using the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are two points on the line.

For line ℓ\ell, the points are (−2,5)(-2, 5) and (4,−7)(4, -7).

Substitute into the slope formula:

m=−7−54−(−2)=−126=−2m = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2Step 2: Use the same slope for the parallel line

Since parallel lines have the same slope, the slope of the line passing through (1,3)(1, 3) is also −2-2.

Step 3: Use point-slope form to write the equation

The point-slope form of the equation of a line is:

y−y1=m(x−x1)y - y_1 = m(x - x_1)where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line.

Substitute the slope m=−2m = -2 and the point (1,3)(1, 3):

y−3=−2(x−1)y - 3 = -2(x - 1)Step 4: Simplify to slope-intercept form

Now, simplify the equation to get it in slope-intercept form y=mx+by = mx + b:

First, distribute the −2-2 on the right-hand side:

y−3=−2x+2y - 3 = -2x + 2Next, add 3 to both sides to solve for yy:

y=−2x+5y = -2x + 5Final Answer:

The equation of the line passing through (1,3)(1, 3) and parallel to line ℓ\ell is:

y=−2x+5y = -2x + 5

Answer 3

Answer 4

Slope m₁ = (- 7 - 5)/(4 + 2) = - 2; m₂ = - 2;

Equation of 2nd line: y = 3 - 2(x - 1); y = - 2x + 5