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ACT Math : How to find out if lines are parallel

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Example Questions

Example Question #191 : Geometry

Which of the following lines is parallel to:

$\displaystyle 4y-12x=2$

Possible Answers:

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

x+4y=10 $\displaystyle x+4y=10$

x-3y=9 $\displaystyle x-3y=9$

$y=\frac{1}{3}x+2$ $\displaystyle y=\frac{1}{3}x+2$

Correct answer:

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

Explanation:

First write the equation in slope intercept form. Add 12x $\displaystyle 12x$ to both sides to get $\displaystyle 4y = 12x + 2$. Now divide both sides by $\displaystyle 4$ to get $\displaystyle y = 3x + 0.5$. The slope of this line is $\displaystyle 3$, so any line that also has a slope of $\displaystyle 3$ would be parallel to it. The correct answer is $x - \frac{1}{3}y = 7$ $\displaystyle x - \frac{1}{3}y = 7$.

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Example Question #1 : How To Find Out If Lines Are Parallel

Which pair of linear equations represent parallel lines?

Possible Answers:

$y=2x-4$ $\displaystyle y=2x-4$

$y=2x+5$ $\displaystyle y=2x+5$

$y=-x+4$ $\displaystyle y=-x+4$

$y=x+6$ $\displaystyle y=x+6$

$y=x-5$ $\displaystyle y=x-5$

$y=3x+5$ $\displaystyle y=3x+5$

$y=x+2$ $\displaystyle y=x+2$

$y=-x+2$ $\displaystyle y=-x+2$

$y=2x+4$ $\displaystyle y=2x+4$

$y=x+4$ $\displaystyle y=x+4$

Correct answer:

$y=2x-4$ $\displaystyle y=2x-4$

$y=2x+5$ $\displaystyle y=2x+5$

Explanation:

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 " $m$ $\displaystyle m$" spot in the linear equation $(y=mx+b)$ $\displaystyle (y=mx+b)$,

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

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Example Question #192 : Geometry

Which of the following equations represents a line that is parallel to the line represented by the equation $\displaystyle 10x-4y=26$?

Possible Answers:

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

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

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

y=5x-1 $\displaystyle y=5x-1$

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

Correct answer:

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

Explanation:

Lines are parallel when their slopes are the same.

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

$\displaystyle -4y=-10+26$

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

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

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

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Example Question #46 : Lines

Line $\displaystyle p$ passes through the points (4, 2) $\displaystyle (4, 2)$ and $\displaystyle (-2, -2)$. Line $\displaystyle Q$ passes through the point (2,3) $\displaystyle (2,3)$ and has a $\textup{y-intercept}$ $\displaystyle \textup{y-intercept}$ of b=0 $\displaystyle b=0$. Are the two lines parallel? If so, what is their slope? If not, what are their slopes?

Possible Answers:

Yes, the lines are parallel with a slope of $m=\frac{2}{3}$ $\displaystyle m=\frac{2}{3}$.

No, the lines are not parallel. Line $\displaystyle P$ has a slope of $m=\frac{2}{3}$ $\displaystyle m=\frac{2}{3}$ and line $\displaystyle Q$ has a slope of m=2 $\displaystyle m=2$.

Yes, the lines are parallel with a slope of m=2 $\displaystyle m=2$.

No, the lines are not parallel. Line $\displaystyle P$ has a slope of m=2 $\displaystyle m=2$ and line $\displaystyle Q$ has slope $m=\frac{2}{3}$ $\displaystyle m=\frac{2}{3}$.

Correct answer:

Yes, the lines are parallel with a slope of $m=\frac{2}{3}$ $\displaystyle m=\frac{2}{3}$.

Explanation:

Finding slope for these two lines is as easy as applying the slope formula to the points each line contains. We know that line $\displaystyle P$ contains points (4, 2) $\displaystyle (4, 2)$ and $\displaystyle (-2, -2)$, so we can apply our slope formula directly (pay attention to negative signs!)

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

Line $\displaystyle Q$ contains point (2,3) $\displaystyle (2,3)$ and, since the y-intercept is always on the vertical axis, (0,0) $\displaystyle (0,0)$. Thus:

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

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

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Example Question #47 : Lines

Line $\displaystyle A$ is described by the equation y=-3x+7 $\displaystyle y=-3x+7$. Line $\displaystyle B$ passes through the points (0,3) $\displaystyle (0,3)$ and (-3,0) $\displaystyle (-3,0)$. Are the two lines parallel? If so, what is their slope? If not, what are their slopes?

Possible Answers:

No, the lines are not parallel. Line $\displaystyle A$ has slope m=-3 $\displaystyle m=-3$ and line $\displaystyle B$ has slope m=1 $\displaystyle m=1$.

Yes, the lines are parallel, and both lines have slope m=-3 $\displaystyle m=-3$.

Yes, the lines are parallel, and both lines have slope m=1 $\displaystyle m=1$.

No, the lines are not parallel. Line $\displaystyle A$ has slope m=1 $\displaystyle m=1$ and line $\displaystyle B$ has slope m=-3 $\displaystyle m=-3$.

Correct answer:

No, the lines are not parallel. Line $\displaystyle A$ has slope m=-3 $\displaystyle m=-3$ and line $\displaystyle B$ has slope m=1 $\displaystyle m=1$.

Explanation:

We are told at the beginning of this problem that line $\displaystyle A$ is described by y=-3x+7 $\displaystyle y=-3x+7$. Since y=mx+b $\displaystyle y=mx+b$ is our slope-intecept form, we can see that m=-3 $\displaystyle m=-3$ for this line. Since parallel lines have equal slopes, we must determine if line $\displaystyle B$ has a slope of $\displaystyle -3$.

Since we know that $\displaystyle B$ passes through points (0,3) $\displaystyle (0,3)$ and (-3,0) $\displaystyle (-3,0)$, we can apply our slope formula:

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

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

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ACT Math : How to find out if lines are parallel

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #191 : Geometry

Example Question #1 : How To Find Out If Lines Are Parallel

Example Question #192 : Geometry

Example Question #46 : Lines

Example Question #47 : Lines

All ACT Math Resources

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