How to find the equation of a perpendicular line - SSAT Upper Level Quantitative (Math)

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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

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

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

A given line is defined by the equation $y=\frac{5}{6}x-9$ . What is the slope of any line that is perpendicular to ?

Answer

For a given line defined by the equation , any line perpendicular to must have a slope that is the negative reciprocal of 's slope , $-\frac{1}{m}$ .

Since in this case $m=\frac{5}{6}$ ,

$-\frac{1}{m}=-\frac{1}{\frac{5}{6}}=\frac{6}{5}$ .

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Question

Given a line defined by the equation $y=\frac{9}{8}x+4$ , which of the following lines is perpendicular to ?

Answer

For a given line defined by the equation , any line perpendicular to must have a slope that is the negative reciprocal of 's slope , $-\frac{1}{m}$ .

In this instance, the slope of line is $m=\frac{9}{8}$ , so $-\frac{1}{m}=-\frac{1}{\frac{9}{8}}=-\frac{8}{9}$ . The only line provided with an equation that has this slope is $y=-\frac{8}{9}x+4$ .

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