Parallel Lines

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Algebra › Parallel Lines

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Which of the following pairs of lines are parallel?

$y=\frac{1}{2}x-8$

Explanation

Lines can be written in the slope-intercept form:

In this form, equals the slope and represents where the line intersects the y-axis.

Parallel lines have the same slope: .

Only one choice contains tow lines with the same slope.

The slope for both lines in this pair is .

Find a line parallel to the line with the equation:

$y=\frac{4}{5}x-\frac{4}{5}$

$y=\frac{4}{5}x+12$

$y=-\frac{4}{5}x-\frac{1}{5}$

$y=\frac{4}{5}$

$y=\frac{5}{4}x+3$

Explanation

Lines can be written in the slope-intercept format:

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

$m=\frac{4}{5}$

Parallel lines share the same slope.

Only one of the choices has a slope of $\frac{4}{5}$ .

$y=\frac{4}{5}x+12$

Find the equation of a line parallel to:

$y=-\frac{3}{4}x-2$

$y=-\frac{3}{4}x+10$

$y=\frac{3}{4}x-2$

$y=\frac{4}{3}x-2$

Explanation

Lines that are parallel have the same slope. Lines can be written in the slope-intercept form:

In this equation, equals the slope and represents the y-intercept.

In the given equation:

$m=-\frac{3}{4}$

Only one of the choices has a slope of $-\frac{3}{4}$ :

$y=-\frac{3}{4}x+10$

Which of the following lines is parallel to a line with the equation:

$y=-\frac{1}{2}x-2$

Explanation

For two lines to be parallel, they must have the same slope.

Lines can be written in the slope-intercept form:

In this equation, equals the slope and represents the y-intercept.

The slope of the given line is:

There is only one line with a slope of .

A line parallel to $\small y=x+6$ passes through the points $\small (1,2)$ and $\small (4,5)$ . Find the equation of this line.

$\small y=x+1$

$\small y=-x+1$

$\small y=x+2$

$\small y=-x-2$

$\small y=2x+1$

Explanation

This problem can be easily solved through using the point-slope formula:

where $\small m$ is the slope and $\small x_1$ and $\small y_1$ signify one of the given points (coordinates).

The problem provides us with two points, so that requirement is fulfilled. We may choose either one when substituting in our values. The only other requirement left is slope. The problem also provides us with this information, but it's not as obviously given. The problem specifies that the line of interest is parallel to $\small y=x+6$ . By definition, lines are parallel when they have the same slope. Given that information, if the two lines are parallel, the line of interest will have the same slope as the given equation: $\small m=1$ . Therefore, we have our required point and the slope. Now we may substitute in all the information and solve for the equation.

Here, we arbitrarily choose the point $\small (1,2)$ .

${\color{Blue} y=x+1}$

Find a line parallel to the line with the equation:

$y=\frac{4}{5}x-\frac{4}{5}$

$y=\frac{4}{5}x+12$

$y=-\frac{4}{5}x-\frac{1}{5}$

$y=\frac{4}{5}$

$y=\frac{5}{4}x+3$

Explanation

Lines can be written in the slope-intercept format:

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

$m=\frac{4}{5}$

Parallel lines share the same slope.

Only one of the choices has a slope of $\frac{4}{5}$ .

$y=\frac{4}{5}x+12$

Find the slope of a line parallel to the line with the equation:

$\frac{1}{2}$

Explanation

Lines can be written in the slope-intercept format:

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

And it has a slope of:

Parallel lines share the same slope.

The parallel line has a slope of .

Find a line parallel to the line that has the equation:

$y=-\frac{1}{20}x-2$

Explanation

Lines can be written using the slope-intercept equation format:

Lines that are parallel have the same slope.

The given line has a slope of:

Only one of the choices also has the same slope and is the correct answer:

Which of the following lines is parallel to

$y = -\frac{1}{2}x -4$

$y = \frac{1}{2}x - 5$

Explanation

When comparing two lines to see if they are parallel, they must have the same slope. To find the slope of a line, we write it in slope-intercept form

where m is the slope.

The original equation

will need to be written in slope-intercept form. To do that, we will divide each term by 4

$\frac{4y}{4} = \frac{-2x}{4} + \frac{8}{4}$

$y = \frac{-1x}{2} + 2$

$y = - \frac{1}{2}x + 2$

Therefore, the slope of the original line is $-\frac{1}{2}$ . A line that is parallel to this line will also have a slope of $-\frac{1}{2}$ .

Therefore, the line

$y = -\frac{1}{2}x -4$

is parallel to the original line.

Find the equation of a line parallel to:

$y=-\frac{3}{4}x-2$

$y=-\frac{3}{4}x+10$

$y=\frac{3}{4}x-2$

$y=\frac{4}{3}x-2$

Explanation

Lines that are parallel have the same slope. Lines can be written in the slope-intercept form:

In this equation, equals the slope and represents the y-intercept.

In the given equation:

$m=-\frac{3}{4}$

Only one of the choices has a slope of $-\frac{3}{4}$ :

$y=-\frac{3}{4}x+10$

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