Parallel Lines - ACT Math

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Question

Which of the following lines is parallel to:

Answer

First write the equation in slope intercept form. Add to both sides to get . Now divide both sides by to get . The slope of this line is , so any line that also has a slope of would be parallel to it. The correct answer is $x - \frac{1}{3}y = 7$ .

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Question

Which pair of linear equations represent parallel lines?

Answer

Parallel lines will always have equal slopes. The slope can be found quickly by observing the equation in slope-intercept form and seeing which number falls in the " $m$ " spot in the linear equation $(y=mx+b)$ ,

We are looking for an answer choice in which both equations have the same $m$ value. Both lines in the correct answer have a slope of 2, therefore they are parallel.

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Question

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

Answer

Lines are parallel when their slopes are the same.

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

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

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

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

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Question

Line passes through the points and . Line passes through the point and has a $\textup{y-intercept}$ of . Are the two lines parallel? If so, what is their slope? If not, what are their slopes?

Answer

Finding slope for these two lines is as easy as applying the slope formula to the points each line contains. We know that line contains points and , so we can apply our slope formula directly (pay attention to negative signs!)

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{2+2}{4+2} = \frac{4}{6} = \frac{2}{3}$ .

Line contains point and, since the y-intercept is always on the vertical axis, . Thus:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{2-0}{3-0} = \frac{2}{3}$

The two lines have the same slope, $m=\frac{2}{3}$ , and are thus identical.

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Question

Line is described by the equation . Line passes through the points and . Are the two lines parallel? If so, what is their slope? If not, what are their slopes?

Answer

We are told at the beginning of this problem that line is described by . Since is our slope-intecept form, we can see that for this line. Since parallel lines have equal slopes, we must determine if line has a slope of .

Since we know that passes through points and , we can apply our slope formula:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{3-0}{0-(-3)} = \frac{3}{3} = 1$

Thus, the slope of line is 1. As the two lines do not have equal slopes, the lines are not parallel.

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Question

If the line through the points (5, –3) and (–2, p) is parallel to the line y = –2_x_ – 3, what is the value of p ?

Answer

Since the lines are parallel, the slopes must be the same. Therefore, (p+3) divided by (_–2–_5) must equal _–_2. 11 is the only choice that makes that equation true. This can be solved by setting up the equation and solving for p, or by plugging in the other answer choices for p.

$\frac{p--3}{-2-5}=\frac{p+3}{-2-5}=-2$

$\frac{p+3}{-7}=-2$

$(\frac{p+3}{-7})-7=-2(-7)$

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Question

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?

Answer

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

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Question

What line is parallel to at ?

Answer

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

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Question

What is the equation of a line that is parallel to and passes through ?

Answer

To solve, we will need to find the slope of the line. We know that it is parallel to the line given by the equation, meaning that the two lines will have equal slopes. Find the slope of the given line by converting the equation to slope-intercept form.

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

The slope of the line will be $-\frac{3}{5}$ . In slope intercept-form, we know that the line will be $y=-\frac{3}{5}x+b$ . Now we can use the given point to find the y-intercept.

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

$4=-\frac{3}{5}(10)+b$

The final equation for the line will be $y=-\frac{3}{5}x+10$ .

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Question

What line is parallel to and passes through the point ?

Answer

Start by converting the original equation to slop-intercept form.

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

The slope of this line is $-\frac{3}{2}$ . A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.

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

$-1=-\frac{3}{2}(2)+b$

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

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

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Question

What is the equation of a line that is parallel to the line $\small y=\frac{1}{2}x+3$ and includes the point ?

Answer

The line parallel to $\small y=\frac{1}{2}x+3$ must have a slope of $\frac{1}{2}$ , giving us the equation $\small y=\frac{1}{2}x+b$ . To solve for b, we can substitute the values for y and x.

$\small 2=(\frac{1}{2})(4)+b$

$\small 2=2+b$

$\small b=0$

Therefore, the equation of the line is $\small y=\frac{1}{2}x$ .

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Question

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

Answer

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.

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

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Question

Which of these formulas could be a formula for a line perpendicular to the line ?

Answer

This is a two-step problem. First, the slope of the original line needs to be found. The slope will be represented by "" when the line is in -intercept form .

$y = \left(\frac{-2}{3}\right)x +\left(\frac{16}{3}\right)$

So the slope of the original line is $-\frac{2}{3}$ . A line with perpendicular slope will have a slope that is the inverse reciprocal of the original. So in this case, the slope would be $\frac{3}{2}$ . The second step is finding which line will give you that slope. For the correct answer, we find the following:

$y= \left(\frac{3}{2}x\right) +\left(\frac{16.5}{6}\right)$

So, the slope is $\frac{3}{2}$ , and this line is perpendicular to the original.

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Question

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

Answer

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 :

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Question

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

Answer

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,

where is the slope. In this particular situation .

Therefore we want to find an equation that has the same value and a different value.

Thus,

is parallel to our equation.

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Question

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

Answer

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.

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Question

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

Answer

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:

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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.

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