Parallel Lines - GMAT Quantitative Reasoning

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Question

Find the equation of a line that is parallel to and passes through the point .

Answer

The parallel line has the equation $\dpi{100} \small 4x-2y=5$ . We can find the slope by putting the equation into slope-intercept form, y = mx + b, where m is the slope and b is the intercept. $\dpi{100} \small 4x-2y=5$ becomes $\dpi{100} \small y=2x-\frac{5}{2}$ , so the slope is 2.

We know that our line must have an equation that looks like $\dpi{100} \small y=2x+b$ . Now we need the intercept. We can solve for b by plugging in the point (4, 1).

1 = 2(4) + b

b = –7

Then the line in question is $\dpi{100} \small y=2x-7$ .

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Question

What is the equation of the line that is parallel to and goes through point ?

Answer

Parallel lines have the same slope. Therefore, the slope of the new line is , as the equation of the original line is ,with slope .

and :

$y-y_{1}=m(x-x_{1})$

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Question

Given:

$\small f(x)=4x+13$

Which of the following is the equation of a line parallel to that has a y-intercept of ?

Answer

Parallel lines have the same slope, so our slope will still be 4. The y-intercept is just the "+b" at the end. In f(x) the y-intercept is 13. In this case, we need to have a y-intercept of -13, so our equation just becomes:

$\small \small f(x)=4x-13$

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Question

Find the equation of the line that is parallel to the and passes through the point .

$\small g(x)=6x+7$

Answer

Two lines are parallel if they have the same slope. The slope of g(x) is 6, so eliminate anything without a slope of 6.

Recall slope intercept form which is .

We know that the line must have an m of 6 and an (x,y) of (8,9). Plug everything in and go from there.

$9=(6\cdot 8)+b$

$\small 9=48+b$

$\small b=9-48=-39$

So we get:

$\small y=6x-39$

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Question

Given the function , which of the following is the equation of a line parallel to and has a -intercept of ?

Answer

Given a line defined by the equation with slope , any line that is parallel to also has a slope of . Since , the slope is and the slope of any line parallel to also has a slope of .

Since also needs to have a -intercept of , then the equation for must be .

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Question

Given the function $f(x)=-\frac{6}{5}x+7$ , which of the following is the equation of a line parallel to and has a -intercept of ?

Answer

Given a line defined by the equation with slope , any line that is parallel to also has a slope of . Since $f(x)=-\frac{6}{5}x+7$ , the slope is $-\frac{6}{5}$ and the slope of any line parallel to also has a slope of $m=-\frac{6}{5}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=-\frac{6}{5}x-2$ .

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Question

Given the function , which of the following is the equation of a line parallel to and has a -intercept of ?

Answer

Given a line defined by the equation with slope , any line that is parallel to also has a slope of . Since , the slope is and the slope of any line parallel to also has a slope of .

Since also needs to have a -intercept of , then the equation for must be .

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Question

What is the slope of the line parallel to ?

Answer

Parallel lines have the same slope. Therefore, rewrite the equation in slope intercept form :

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Question

Find the slope of any line parallel to the following function.

Answer

We need to rearrange this equation to get into form.

Begin by adding 6 to both sides to get

Next, divide both sides by 4 to get our slope

$y= \frac{3}{4}x+\frac{18}{4}$

So our slope, m, is equal to 3/4. Therefore, any line with the slope 3/4 will be parallel to the original function.

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Question

A given line is defined by the equation $y=\frac{1}{3}x+9$ . What is the slope of any line parallel to this line?

Answer

Any line that is parallel to a line has a slope that is equal to the slope . Given $y=\frac{1}{3}x+9$ , $m=\frac{1}{3}$ and therefore any line parallel to the given line must have a slope of $\frac{1}{3}$ .

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Question

A given line is defined by the equation . What is the slope of any line parallel to this line?

Answer

Any line that is parallel to a line has a slope that is equal to the slope . Given , and therefore any line parallel to the given line must have a slope of .

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Question

A given line is defined by the equation $y=\frac{4}{3}x+12$ . What is the slope of any line parallel to this line?

Answer

Any line that is parallel to a line has a slope that is equal to the slope . Given $y=\frac{4}{3}x+12$ , $m=\frac{4}{3}$ and therefore any line parallel to the given line must have a slope of $\frac{4}{3}$ .

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Question

Determine whether and $y+8=-\frac{2}{3}x +2$ are parallel lines.

Answer

Parallel lines have the same slope. Therefore, we need to find the slope once both equations are in slope intercept form :

$y=-\frac{2}{3}x+\frac{10}{3}$

$m=-\frac{2}{3}$

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

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

$y=-\frac{2}{3}x-6$

$m=-\frac{2}{3}$

The lines are parallel because the slopes are the same.

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Question

Which of the following pairs of lines are parallel?

Answer

Two lines are parallel if they have the same slope. Let's go through the answer choices.

1. and : Both slopes are 3, parallel.

2. and : Slopes are 2 and 3, not parallel.

3. and : We need to put these two equations into the form of to find the slope, .

$3x - 2y=5\Rightarrow -2y=5-3x\Rightarrow y=3x/2-5/2$ , so the first slope is .

$2x+3y=4\Rightarrow 3y=4-2x\Rightarrow y=-2x/3 + 4/3$ , so the second slope is . These lines are perpendicular, not parallel.

4. and : Slopes are undefined and 0, respectively, so not parallel.

5. and : Let's again put these into the form of .

$5x+4y=1\Rightarrow 4y=1-5x\Rightarrow y=-5x/4 + 1/4$ , so the first slope is .

$4x+5y=2\Rightarrow 5y=2-4x\Rightarrow y=-4x/5+2/5$ , so the second slope is . The lines are not parallel.

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Question

Three lines - one green, one blue, one red - are drawn on the coordinate axes. They have the following characteristics:

The green line has -intercept and -intercept .

The blue line has -intercept and -intercept .

The red line has -intercept and -intercept .

Which of these lines are parallel to each other?

Answer

Use the intercepts to find the slopes of the lines.

Green: $m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{0-(-5)}{2-0} = \frac{5}{2}$

Blue: $m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{5-0}{0-2} =- \frac{5}{2}$

Red: $m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{0-(-5)}{-2-0} =- \frac{5}{2}$

The blue and red lines have the same slope; the green line has a different slope from the other two. That means only the blue and red lines are parallel.

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Question

Which of the following lines is parallel to ?

Answer

Two lines are parallel to each other if their slopes have the same value. Since the slope of is , is the only other line provided that has the same slope.

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Question

Determine whether and $y=-\frac{1}{7}x+9$ are parallel lines.

Answer

By definition, lines that are parallel to each other must have the same slope. has a slope of $m_{1}=7$ and $y=-\frac{1}{7}x+9$ has a slope of $m_{2}=-\frac{1}{7}$ , therefore they are not parallel because their slopes are not the same.

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Question

What is the slope of a line that is parallel to $y=\frac{1}{9}x+15$ ?

Answer

Two lines are parallel if their slopes have the same value. Since $y=\frac{1}{9}x+15$ has a slope of $\frac{1}{9}$ , any line parallel to it must also have a slope of $\frac{1}{9}$ .

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Question

Tell whether the lines described by the following are parallel.

g(x) is a linear equation which passes through the points and .

Answer

Tell whether the lines described by the following are parallel.

g(x) is a linear equation which passes through the points and

Begin by recalling that lines are parallel if their slopes are the same and they have different y-intercepts.

Let's find the slope of g(x)

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{34-20}{-17-43}=\frac{14}{-60}=\frac{-7}{30}$

Note that this slope is not the same as f(x), so we do not have parallel lines!

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