How to find the equation of a perpendicular line - ACT Math

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Question

What line is perpendicular to x + 3_y_ = 6 and travels through point (1,5)?

Answer

Convert the equation to slope intercept form to get y = –1/3_x_ + 2. The old slope is –1/3 and the new slope is 3. Perpendicular slopes must be opposite reciprocals of each other: _m_1 * _m_2 = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3_x_ + 2

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Question

What is the equation of a line that runs perpendicular to the line 2_x_ + y = 5 and passes through the point (2,7)?

Answer

First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2_x_ +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2_x_ + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½_x_ + 6. Rearranged, it is –x/2 + y = 6.

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Question

Which of the following equations represents a line that goes through the point and is perpendicular to the line ?

Answer

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

Transform the original equation to find its slope.

First, subtract from both sides of the equation.

Simplify and rearrange.

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:

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

Add 4 to both sides of the equation.

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

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Question

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?

Answer

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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Question

Which line below is perpendicular to ?

Answer

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 .

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

According to our 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 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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Question

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

Answer

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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Question

What is the equation for the line that is perpendicular to through point ?

Answer

Perpendicular slopes are opposite reciprocals.

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

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

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.

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

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

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Question

What line is perpendicular to and passes through ?

Answer

Convert the given equation to slope-intercept form.

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

The slope of this line is $-\frac{3}{5}$ . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is $\frac{5}{3}$ .

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

$2 = \frac{5}{3}(3) + b$

So the equation of the perpendicular line is $y = \frac{5}{3}x - 3$ .

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Question

Which of the following is possibly a line perpendicular to ?

Answer

To start, begin by dividing everything by , this will get your equation into the format . 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 . If you solve this for , you will get:

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

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Question

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

?

Answer

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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Question

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

Answer

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,

.

In our case we see that

therefore, .

Since

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

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

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Question

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 .

Answer

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 , , we can use point slope form to find an initial equation for our line.

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

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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