Algebra 1 5.6 Homework Assignment E7 And E8 VERIFIED

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How to Solve Parallel and Perpendicular Lines Problems in Algebra 1

In this article, we will learn how to write equations of parallel and perpendicular lines in slope-intercept form, using point-slope form and some basic algebra rules. We will use some examples from the algebra 1 5.6 homework assignment e7 and e8[^1^] [^2^] [^3^] to illustrate the steps.

Parallel Lines

Two lines are parallel if they have the same slope. To write an equation of a line that is parallel to a given line and passes through a given point, we can use the following steps:

Find the slope of the given line. If the equation is in slope-intercept form (y = mx + b), then the slope is m. If the equation is in standard form (Ax + By = C), then the slope is -A/B. If the equation is in point-slope form (y - y1 = m(x - x1)), then the slope is m.

Use the same slope for the parallel line. Write an equation in point-slope form using the given point and the slope. For example, if the slope is m and the point is (x1, y1), then the equation is y - y1 = m(x - x1).

Rearrange the equation in point-slope form to get it in slope-intercept form. To do this, we need to isolate y on one side of the equation by adding or subtracting terms from both sides. For example, if the equation is y - y1 = m(x - x1), then we can add y1 to both sides and get y = m(x - x1) + y1.

Example 1

Write an equation of the line that is parallel to y = x + 5 and passes through (-1, -1).

The slope of the given line is 1, since it is in slope-intercept form and m = 1.

The parallel line has the same slope, so we write an equation in point-slope form using the point (-1, -1) and the slope 1: y - (-1) = 1(x - (-1)).

We simplify and rearrange the equation in point-slope form to get it in slope-intercept form: y + 1 = x + 1; y = x.

The equation of the parallel line is y = x.

Perpendicular Lines

Two lines are perpendicular if their slopes are negative reciprocals of each other. That means that if one line has a slope of m, then the other line has a slope of -1/m. To write an equation of a line that is perpendicular to a given line and passes through a given point, we can use similar steps as for parallel lines:

Find the negative reciprocal of the slope for the perpendicular line. For example, if the slope of the given line is 2, then the slope of the perpendicular line is -1/2.

Write an equation in point-slope form using the given point and the perpendicular slope. For example, if the perpendicular slope is -1/2 and

the point is (x1, y1), then

the equation

(

Rearrange the equation in point-slope form to get it in slope-intercept form. To do this, we need to isolate y on one side of the equation by adding or subtracting terms from both sides. For example, if the equation is y - y1 = -1/m(x - x1), then we can add y1 to both sides and get y = -1/m(x - x1) + y1.

Example 2

Write an equation of the line that is perpendicular to y = 3x - 1 and passes through (1, -3).

The slope of the given line is 3, since it is in slope-intercept form and m = 3.

The negative reciprocal of the slope for the perpendicular line is -1/3. For example, if the slope of the given line is 3, then the slope of the perpendicular line is -1/3.

Write an equation in point-slope form using the point (1, -3) and the perpendicular slope -1/3: y - (-3) = -1/3(x - 1).

Rearrange the equation in point-slope form to get it in slope-intercept form: y + 3 = -1/3(x - 1); y = -x/3 - 8/3.

The equation of the perpendicular line is y = -x/3 - 8/3.

Practice Problems

Now that we have learned how to write equations of parallel and perpendicular lines, let's try some practice problems from the algebra 1 5.6 homework assignment e7 and e8[^1^] [^2^] [^3^]. You can check your answers with the answer key at the end of this article.

Write an equation of the line that is parallel to y = -3x + 1 and passes through (2, 4).

Write an equation of the line that is parallel to y = 2x â 11 and passes through (3, 4).

Write an equation of the line that is parallel to y = 1/4 x â 6 and passes through (3, 3).

Write an equation of the line that is parallel to y = 1/2 x and passes through (8, -10).

Write an equation of the line that is parallel to y = 1/3 x + 4 and passes through (-4, -4).

Write an equation of the line that is parallel to the line in the graph below and passes through (-2, -4).

Write an equation of the line that is parallel to the line in the graph below and passes through (0, 0).

Write an equation of the line that is perpendicular to y = x + 2 and passes through (3, 0).

Write an equation of the line that is perpendicular to y = -2x + 8 and passes through (-3, 1).

Write an equation of the line that is perpendicular to y = -1/2 x + 4 and passes through (8, 5).

Write an equation of the line that is perpendicular to y = 7/8 x and passes through (0, 3).

Write an equation of the line that is perpendicular to y = -1/2 x + 4 and passes through (8, 5).

Write an equation of the line that is perpendicular to

the

line

the

graph

below

and

passes

through

(4,

-2).

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