ACT Math - How to find the equation of a parallel line

Which of the following is the equation of a line parallel to the line given by the equation:

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

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

Explanation

Parallel lines have the same slope and different y-intercepts. If their y-intercepts and slopes are the same they are the same line, and therefore not parallel. Thus the only one that fits the description is:

Which of the following equations represents a line that is parallel to the line represented by the equation ?

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

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

$y=\frac{2}{5}x+2$

$y=-\frac{2}{5}x+3$

Explanation

Lines are parallel when their slopes are the same.

First, we need to place the given equation in the slope-intercept form.

Subtract from both sides of the equation.

Simplify.

Divide both sides of the equation by .

$\frac{-4y}{-4}=\frac{-10}{-4}+\frac{26}{-4}$

Simplify.

$y=\frac{10}{4}x-\frac{26}{4}$

Reduce.

$y=\frac{5}{2}x-\frac{13}{2}$

Because the given line has the slope of $\frac{5}{2}$ , the line parallel to it must also have the same slope.

There is a line defined by the equation below:

There is a second line that passes through the point and is parallel to the line given above. What is the equation of this second line?

$y=-\frac{3}{4}x+2.75$

$y=-\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.75$

$y=\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.625$

Explanation

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = _–_3x + 12

y = –(3/4)x + 3

slope = _–_3/4

We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = _–_3/4(1) + b

2 = _–_3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

What is the equation of a line parallel to the line given by the equation:
?

$y = -\frac{1}{9}x + -1$

$y = \frac{1}{9}x - 5$

$y = x + \frac{1}{9}$

Explanation

Parallel lines have the same slope and differing y-intercepts. Since is the only equation with the same slope, and the y-intercept is different, this is the equation of the parallel line.

A line passes through and . Give the equation, in slope-intercept form, of a parallel line that passes through .

$y=\frac{2}{3}x + \frac{16}{3}$

$y=\frac{2}{3}x+\frac{8}{3}$

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

$y=-\frac{2}{3}x-\frac{11}{3}$

Explanation

The first step in solving for a parallel line is to find the slope of the original line. In this case, $m=\frac{(8-2)}{(13-4)} = \frac{6}{9} = \frac{2}{3}$ .

Our new line has an equal slope (the definition of parallel) and passes through , so substituting into our slope-intercept equation gives us:

---> $4=\frac{2}{3}(-2) +b$ ---> $\frac{12}{3} + \frac{4}{3} = b = \frac{16}{3}$ .

Thus, the slope intercept equation for our parallel line is $y=\frac{2}{3}x + \frac{16}{3}$ .

Which of the following is a line that is parallel to the line defined by the equation ?

Explanation

Since parallel lines have equal slopes, you should find the slope of the line given to you. The easiest way to do this is to solve the equation so that its form is . represents the slope.

Take your equation:

First, subract from both sides:

Next, subtract from both sides:

Finally, divide by :

$y = \frac{49}{3} - 2x$ , which is the same as $y = - 2x + \frac{49}{3}$

Thus, your slope is .

Among the options provided only is parallel. Solve this equation as well for form.

First, subtract from both sides:

Then, divide by :

What line is parallel to , and passes through the point ?

$y = \frac{-5}{3}x + 6$

$y = \frac{3}{5}x - 2$

$y = \frac{1}{3}x - 3$

$y = \frac{-2}{3}x + 5$

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

Explanation

Converting the given line to slope-intercept form we get the following equation:

$y = \frac{-5}{3}x + \frac{8}{3}$

For parallel lines, the slopes must be equal, so the slope of the new line must also be $\frac{-5}{3}$ . We can plug the new slope and the given point into the slope-intercept form to solve for the y-intercept of the new line.

$-4 = \frac{-5}{3}(6) + b$

Use the y-intercept in the slope-intercept equation to find the final answer.

$y = \frac{-5}{3}x + 6$

What line is parallel to at ?

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

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

$y=-\frac{3}{2}x+8$

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

None of the answers are correct

Explanation

Find the slope of the given line: (slope intercept form)

$y=\frac{-2}{3}x+2$ therefore the slope is $\frac{-2}{3}$

Parallel lines have the same slope, so now we need to find the equation of a line with slope $\frac{-2}{3}$ and going through point by substituting values into the point-slope formula.

$2=\frac{-2}{3}(3)+b$

So,

Thus, the new equation is $y=\frac{-2}{3}x+4$

What is the equation of the line passing through and which is parallel to $y=\frac{1}{3}x+7$ ?

$y=\frac{1}{3}x$

$y=\frac{7}{3}x +1$

$y=\frac{1}{3}x-7$

$\textup{There is not enough information to give an answer.}$

Explanation

Ordinarily, we would use point-slope form, , to construct a proper parallel from a point and a slope. However, in this case, we have been given a y-intercept by the problem itself, so sticking with slope-intercept form is easiest.

If the new line passes through , then we know the y-intercept is . Since slope is equal in parallel lines, and the slope of our comparison line is $m=\frac{1}{3}$ , we know the slope of our new line is $m=\frac{1}{3}$ .

Thus, $y=mx+b = \frac{1}{3}x +0 = \frac{1}{3}x$ .

The correct answer is, $y=\frac{1}{3}x$ .

Which of the following answer choices gives the equation of a line parallel to the line:

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

Explanation

Parallel lines have the same slope but different y-intercepts. When the equations of two lines are the same they have infinitely many points in common, whereas parallel lines have no points in common.

Our equation is given in slope-intercept form,