Parallel Lines

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GMAT Quantitative › Parallel Lines

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

Yes, because the slopes are the same.

Yes, because the slopes are negative reciprocals of each other.

No, because the slopes are not similar.

No, because the slopes are not negative reciprocals of each other.

Explanation

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.

Tell whether the lines described by the following are parallel.

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

They are not parallel, because they have different slopes.

There are not parallel, they are the same line.

They are parallel, because they have the same slope.

They are parallel, because their slopes are opposite reciprocals of each other.

Explanation

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!

What is the slope of the line parallel to ?

Explanation

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

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?

$\frac{1}{3}$

$-\frac{1}{3}$

$-\frac{2}{3}$

Explanation

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 $\frac{1}{3}$ .

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

$g(x)=\frac{1}{7}x+8$

$g(x)=-\frac{1}{7}x+8$

Explanation

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 .

Which of the following lines is parallel to ?

$y=\frac{1}{3}x-5$

$y=-\frac{1}{3}x-5$

Two of the answers provided are correct.

Explanation

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.

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

No, because the slopes are not the same.

Yes, because the slopes are the same.

No, because the slopes are not negative reciprocals of each other.

Yes, because the slopes are negative reciprocals of each other.

Cannot determine.

Explanation

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.

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

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

$\small y=6x-39$

$\small \small y=6x+39$

$\small \small y=-6x-39$

$\small \small y=\frac{1}{6}x-39$

$\small \small \small y=-\frac{1}{6}x-39$

Explanation

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:

$\small y=6x-39$

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

$\dpi{100} \small y=2x-7$

$\dpi{100} \small y=-2x-7$

$\dpi{100} \small y=2x+7$

$\dpi{100} \small y=-2x+7$

$\dpi{100} \small y< 2x-7$ None of the answers are correct.

Explanation

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 $\dpi{100} \small y=2x-7$ .

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 ?

$g(x)=-\frac{6}{5}x-2$

$g(x)=\frac{6}{5}x-2$

$g(x)=-\frac{5}{6}x-2$

$g(x)=\frac{5}{6}x-2$

$g(x)=-\frac{6}{5}x+2$

Explanation

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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