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ACT Math : Lines

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Example Questions

Example Question #72 : Coordinate Geometry

Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?

Possible Answers:

3x + 2y = 4

2x – y = 3

4x – 3y = 4

2x + y = 3

x – y = 3

Correct answer:

2x – y = 3

Explanation:

The slope of m is equal to ^y2-y1/_x2-x1=^2-4/_5-1= ^-1/₂

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be –1:

(slope of p) * (^-1/₂) = -1

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x – y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

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Example Question #1 : How To Find The Equation Of A Perpendicular Line

What is the equation for the line that is perpendicular to 4x - 3y = 6 through point (4, 6) ?

Possible Answers:

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

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

$y = \frac{-3}{4}x +9$

$y = \frac{4}{3}x + 6$

$y = \frac{4}{3}x - 5$

Correct answer:

$y = \frac{-3}{4}x +9$

Explanation:

Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

4x - 3y = 6

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

The slope of the given line is $m = \frac{4}{3}$ and the perpendicular slope is $m = \frac{-3}{4}$ .

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

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

b = 9

Using this y-intercept in slope-intercept form, we get out final equation: $y = \frac{-3}{4}x + 9$ .

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Example Question #2 : How To Find The Equation Of A Perpendicular Line

Which line below is perpendicular to 5x+6y=18 ?

Possible Answers:

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

$y=\frac{5}{6}x+2$

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

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

$y=-\frac{6}{5}x+8$

Correct answer:

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

Explanation:

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or y=mx+b .

5x+6y=18

6y=-5x+18

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

According to our y=mx+b formula, our slope for the original line is $-\frac{5}{6}$ . We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of $-\frac{5}{6}$ is $\frac{6}{5}$ . Flip the original and multiply it by .

Our answer will have a slope of $\frac{6}{5}$ . Search the answer choices for $\frac{6}{5}$ in the position of the y=mx+b equation.

$\dpi{100} y = \frac{6}{5}x + 3$ is our answer.

(As an aside, the negative reciprocal of 4 is $-\frac{1}{4}$ . Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

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Example Question #1 : How To Find The Equation Of A Perpendicular Line

If a line has an equation of $2y=3x+3$ , what is the slope of a line that is perpendicular to the line?

Possible Answers:

$-\frac{3}{2}$

$-\frac{2}{3}$

$\frac{3}{2}$

$-2$

$3$

Correct answer:

$-\frac{2}{3}$

Explanation:

Putting the first equation in slope-intercept form yields $y=\frac{3}{2}x+\frac{3}{2}$ .

A perpendicular line has a slope that is the negative inverse. In this case, $-\frac{2}{3}$ .

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Example Question #3 : How To Find The Equation Of A Perpendicular Line

Which of the following is possibly a line perpendicular to 9y=3x + 22 ?

Possible Answers:

4y - 12x = 31

3y + 9x = 54

3x + 2y =34

5x - 2y = 18

4y - 8x = 2

Correct answer:

3y + 9x = 54

Explanation:

To start, begin by dividing everything by , this will get your equation into the format y=mx+b . This gives you:

$y=\frac{1}{3}x+\frac{22}{9}$

Now, recall that the slope of a perpendicular line is the opposite and reciprocal slope to its mutually perpendicular line. Thus, if our slope is $\frac{1}{3}$ , then the perpendicular line's slope must be . Thus, we need to look at our answers to determine which equation has a slope of . Among the options given, the only one that matches this is 3y + 9x = 54 . If you solve this for , you will get:

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

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Example Question #1 : How To Find The Equation Of A Perpendicular Line

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

y = -3x +9 ?

Possible Answers:

y = -3x + 6

y = x + 4

y = 3x - 9

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

y = -3x - 9

Correct answer:

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

Explanation:

For two lines to be perpendicular their slopes must have a product of .
$-3*\frac{1}{3} = -1$ and so we see the correct answer is given by $y = \frac{1}{3}x + 2$

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Example Question #3 : How To Find The Equation Of A Perpendicular Line

What is the equation of a line perpendicular to the line defined by the equaiton:

y = -9x + 2

Possible Answers:

y = 9x - 13

$y = -\frac{1}{9}x -2$

y = -9x + 3

y = -9x + 1

$y = \frac{1}{9}x +6$

Correct answer:

$y = \frac{1}{9}x +6$

Explanation:

Perpendicular lines have slopes whose product is .

Looking at our equations we can see that it is in slope-intercept form where the m value represents the slope of the line,

y=mx+b .

In our case we see that

y = -9x + 2 therefore, m=-9 .

Since

$-9*\frac{1}{9} = -1$ we see the only possible answer is

$y = \frac{1}{9}x +6$ .

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Example Question #11 : How To Find The Equation Of A Perpendicular Line

Line follows the equation $y=\frac{3}{4}x -2$ .

State the equation, in slope-intercept form, of line , which is perpendicular to line and intersects it at point (24, 18) .

Possible Answers:

$y=-\frac{4}{3}x + 50$

$y=-\frac{4}{3}x + 14$

$y=\frac{4}{3}x + 42$

$y=\frac{3}{4}x -6$

Correct answer:

$y=-\frac{4}{3}x + 50$

Explanation:

Since line follows the equation $y=\frac{3}{4}x -2$ , we can surmise its slope is $m=\frac{3}{4}$ . Thus, it follows that any line perpendicular to will have a slope of $n = -\frac{1}{m} = -\frac{1}{(\frac{3}{4})} = -\frac{4}{3}$ .

Since we also know at least one point on line , (24, 18) , we can use point slope form to find an initial equation for our line.

y-y_1=m(x-x_1) ----> $y-18=-\frac{4}{3}(x-24)$ .

Next, we can simplify to reach point-slope form.

$y-18=-\frac{4}{3}(x-24)$ ----> $y=-\frac{4}{3}x + 50$ .

Thus, line in slope-intercept form is $y=-\frac{4}{3}x + 50$ .

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Example Question #21 : How To Find The Equation Of A Perpendicular Line

Which of the following equations represents a line that goes through the point (-2, 6) and is perpendicular to the line 3x + 6y = 12 ?

Possible Answers:

y = 2x + 6

y = -2x + 10

y = 2x + 10

y = 2x + 2

y = -2x + 2

Correct answer:

y = 2x + 10

Explanation:

In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:

y = mx + b

Transform the original equation to find its slope.

3x + 6y = 12

First, subtract from both sides of the equation.

3x-3x + 6y = 12-3x

Simplify and rearrange.

6y = -3x + 12

Next, divide both sides of the equation by 6.

$\frac{6y}{6} = -\frac{3x}{6} + \frac{12}{6}$

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

The slope of our first line is equal to $-\frac{1}{2}$ . Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following:

$-\frac{1}{x}$

Let's calculate the opposite reciprocal of our slope:

$-\frac{1}{2}\rightarrow -\left (-\frac{2}{1} \right )=2$

The slope of our line is equal to 2. We now have the following partial equation:

y = 2x + b

We are missing the y-intercept, . Substitute the x- and y-values in the given point (-2, 6) to solve for the missing y-intercept.

6 = 2(-2) + b

6 = -4 + b

Add 4 to both sides of the equation.

6+4 = -4+4 + b

10 = b

Substitute this value into our partial equation to construct the equation of our line:

y = 2x + 10

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Example Question #1 : How To Find The Slope Of A Perpendicular Line

What is the slope of any line perpendicular to 2y = 4x +3 ?

Possible Answers:

– ½

– 4

Correct answer:

– ½

Explanation:

First, we must solve the equation for y to determine the slope: y = 2x + ³/₂

By looking at the coefficient in front of x, we know that the slope of this line has a value of 2. To fine the slope of any line perpendicular to this one, we take the negative reciprocal of it:

slope = m , perpendicular slope = – ¹/_m

slope = 2 , perpendicular slope = – ¹/₂

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ACT Math : Lines

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #72 : Coordinate Geometry

Example Question #1 : How To Find The Equation Of A Perpendicular Line

Example Question #2 : How To Find The Equation Of A Perpendicular Line

Example Question #1 : How To Find The Equation Of A Perpendicular Line

Example Question #3 : How To Find The Equation Of A Perpendicular Line

Example Question #1 : How To Find The Equation Of A Perpendicular Line

Example Question #3 : How To Find The Equation Of A Perpendicular Line

Example Question #11 : How To Find The Equation Of A Perpendicular Line

Example Question #21 : How To Find The Equation Of A Perpendicular Line

Example Question #1 : How To Find The Slope Of A Perpendicular Line

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