### All ACT Math Resources

## Example Questions

### Example Question #1 : Coordinate Geometry

There is a line defined by the equation below:

There is a second line that passes through the point and is parallel to the line given above. What is the equation of this second line?

**Possible Answers:**

**Correct answer:**

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = *–*3x + 12

y = *–*(3/4)x + 3

slope = *–*3/4

We know that the second line will also have a slope of *–*3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = *–*3/4(1) + b

2 = *–*3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = *–*(3/4)x + 2.75

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

What is the equation of a line that is parallel to and passes through ?

**Possible Answers:**

**Correct answer:**

To solve, we will need to find the slope of the line. We know that it is parallel to the line given by the equation, meaning that the two lines will have equal slopes. Find the slope of the given line by converting the equation to slope-intercept form.

The slope of the line will be . In slope intercept-form, we know that the line will be . Now we can use the given point to find the y-intercept.

The final equation for the line will be .

### Example Question #3 : Coordinate Geometry

What line is parallel to and passes through the point ?

**Possible Answers:**

**Correct answer:**

Start by converting the original equation to slop-intercept form.

The slope of this line is . A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.

Plug the y-intercept into the slope-intercept equation to get the final answer.

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

What is the equation of a line that is parallel to the line and includes the point ?

**Possible Answers:**

**Correct answer:**

The line parallel to must have a slope of , giving us the equation . To solve for *b*, we can substitute the values for *y* and *x*.

Therefore, the equation of the line is .

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

What line is parallel to , and passes through the point ?

**Possible Answers:**

**Correct answer:**

Converting the given line to slope-intercept form we get the following equation:

For parallel lines, the slopes must be equal, so the slope of the new line must also be . We can plug the new slope and the given point into the slope-intercept form to solve for the y-intercept of the new line.

Use the y-intercept in the slope-intercept equation to find the final answer.

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

What line is parallel to at ?

**Possible Answers:**

None of the answers are correct

**Correct answer:**

Find the slope of the given line: (slope intercept form)

therefore the slope is

Parallel lines have the same slope, so now we need to find the equation of a line with slope and going through point by substituting values into the point-slope formula.

So,

Thus, the new equation is

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

If the line through the points (5, –3) and (–2, *p*) is parallel to the line *y* = –2*x* – 3, what is the value of *p* ?

**Possible Answers:**

0

*–*10

11

4

*–*17

**Correct answer:**

11

Since the lines are parallel, the slopes must be the same. Therefore, (p+3) divided by (*–*2*–*5) must equal *–*2. 11 is the only choice that makes that equation true. This can be solved by setting up the equation and solving for p, or by plugging in the other answer choices for p.

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

Which of these formulas could be a formula for a line perpendicular to the line ?

**Possible Answers:**

**Correct answer:**

This is a two-step problem. First, the slope of the original line needs to be found. The slope will be represented by "" when the line is in -intercept form .

So the slope of the original line is . A line with perpendicular slope will have a slope that is the inverse reciprocal of the original. So in this case, the slope would be . The second step is finding which line will give you that slope. For the correct answer, we find the following:

So, the slope is , and this line is perpendicular to the original.

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

Which of the following is a line that is parallel to the line defined by the equation ?

**Possible Answers:**

**Correct answer:**

Since parallel lines have equal slopes, you should find the slope of the line given to you. The easiest way to do this is to solve the equation so that its form is . represents the slope.

Take your equation:

First, subract from both sides:

Next, subtract from both sides:

Finally, divide by :

, which is the same as

Thus, your slope is .

Among the options provided only is parallel. Solve this equation as well for form.

First, subtract from both sides:

Then, divide by :

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

Which of the following answer choices gives the equation of a line parallel to the line:

**Possible Answers:**

**Correct answer:**

Parallel lines have the same slope but different y-intercepts. When the equations of two lines are the same they have infinitely many points in common, whereas parallel lines have no points in common.

Our equation is given in slope-intercept form,

where is the slope. In this particular situation .

Therefore we want to find an equation that has the same value and a different value.

Thus,

is parallel to our equation.