### All ACT Math Resources

## Example Questions

### Example Question #45 : Lines

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

**Possible Answers:**

y = -2x + 2

y = 2x + 10

y = 2x + 2

y = 2x + 6

y = -2x + 10

**Correct answer:**

y = 2x + 10

We must must transform the standard form equation 3x + 6y = 12 into a slope-intercept form equation (y = mx + b) to find its slope.

3x + 6y = 12 (Subtract 3x on both sides.)

6y = -3x + 12 (Divide both sides by 6.)

y = -3/6x + 12/6

y = -1/2x + 2

The slope of our first line is equal to -1/2. Perpendicular lines have negative reciprocal slopes, so if the slope of one is x, the slope of the other is -1/x.

The negative reciprocal of -1/2 is equal to 2, so that is the slope of our line.

We now have y = 2x + b. We plug in the point (-2, 6) and solve for b.

6 = 2(-2) + b

6 = -4 + b (Add 4 on both sides)

10 = b

The equation of our new line is **y = 2x + 10**

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

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

**Possible Answers:**

*y* = 6*x* – 3

*y* = 2/3*x* + 6

*y* = 2*x* + 1

*y* = –1/3*x* – 4

*y* = 3*x* + 2

**Correct answer:**

*y* = 3*x* + 2

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

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

What line is perpendicular to and passes through ?

**Possible Answers:**

**Correct answer:**

Convert the given equation to slope-intercept form.

The slope of this line is . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is .

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

So the equation of the perpendicular line is .

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

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

**Possible Answers:**

*x*/2 – *y* = 6

2*x* – *y* = 6

2*x* + *y* = 7

–*x*/2 + *y* = 6

*x*/2 + *y* = 5

**Correct answer:**

–*x*/2 + *y* = 6

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.

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

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

x **–** y = 3

4x **–** 3y = 4

2x + y = 3

**Correct answer:**

2x **–** y = 3

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.

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

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

**Possible Answers:**

**Correct answer:**

Perpendicular slopes are opposite reciprocals.

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

The slope of the given line is and the perpendicular slope is .

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.

Using this y-intercept in slope-intercept form, we get out final equation: .

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

Which line below is perpendicular to ?

**Possible Answers:**

**Correct 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 .

According to our formula, our slope for the original line is . We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of is . Flip the original and multiply it by .

Our answer will have a slope of . Search the answer choices for in the position of the equation.

is our answer.

(As an aside, the negative reciprocal of 4 is . 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.)

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

If a line has an equation of , what is the slope of a line that is perpendicular to the line?

**Possible Answers:**

**Correct answer:**

Putting the first equation in slope-intercept form yields .

A perpendicular line has a slope that is the negative inverse. In this case, .

### Example Question #5 : How To Find The Equation Of A Perpendicular Line

Which of the following is possibly a line perpendicular to ?

**Possible Answers:**

**Correct answer:**

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

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 , 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:

### Example Question #3 : 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:

?

**Possible Answers:**

**Correct answer:**

For two lines to be perpendicular their slopes must have a product of .

and so we see the correct answer is given by

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