ACT Math : How to find the equation of a line

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #247 : Geometry

What line goes through the points (0, 6) and (4, 0)?

Possible Answers:

y = 2/3x –6

y = –3/2 – 3

y = 2/3 + 5

y = –3/2x + 6

y = 1/5x + 3

Correct answer:

y = –3/2x + 6

Explanation:

P1 (0, 6) and P2 (4, 0)

First, calculate the slope:  m = rise ÷ run = (y2 – y1)/(x– x1), so m = –3/2

Second, plug the slope and one point into the slope-intercept formula: 

y = mx + b, so 0 = –3/2(4) + b and b = 6

Thus, y = –3/2x + 6

Example Question #1 : Other Lines

What line goes through the points (1, 3) and (3, 6)?

Possible Answers:

3x + 5y = 2

–3x + 2y = 3

4x – 5y = 4

2x – 3y = 5

–2x + 2y = 3

Correct answer:

–3x + 2y = 3

Explanation:

If P1(1, 3) and P2(3, 6), then calculate the slope by m = rise/run = (y2 – y1)/(x2 – x1) = 3/2

Use the slope and one point to calculate the intercept using y = mx + b

Then convert the slope-intercept form into standard form.

Example Question #2 : Other Lines

What is the slope-intercept form of \dpi{100} \small 8x-2y-12=0?

Possible Answers:

\dpi{100} \small y=4x+6

\dpi{100} \small y=-2x+3

\dpi{100} \small y=-4x+6

\dpi{100} \small y=2x-3

\dpi{100} \small y=4x-6

Correct answer:

\dpi{100} \small y=4x-6

Explanation:

The slope intercept form states that \dpi{100} \small y=mx+b. In order to convert the equation to the slope intercept form, isolate \dpi{100} \small y on the left side:

\dpi{100} \small 8x-2y=12

\dpi{100} \small -2y=-8x+12

\dpi{100} \small y=4x-6

Example Question #3 : Other Lines

A line is defined by the following equation:

What is the slope of that line?

Possible Answers:

Correct answer:

Explanation:

The equation of a line is

y=mx + b where m is the slope

Rearrange the equation to match this:

7x + 28y = 84

28y = -7x + 84

y = -(7/28)x + 84/28

y = -(1/4)x + 3

m = -1/4

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

If the coordinates (3, 14) and (5, 15) are on the same line, what is the equation of the line?

Possible Answers:

Correct answer:

Explanation:

First solve for the slope of the line, m using y=mx+b

m = (y2 – y1) / (x2 – x1)

= (15  14) / (5 3)

= (1 )/( 8)

=1/8

y = (1/8)x + b

Now, choose one of the coordinates and solve for b:

14 = (1/8)3 + b

14 = 3/8 + b

b = 14 + (3/8)

b = 14.375

y = (1/8)x + 14.375

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

What is the equation of a line that passes through coordinates \dpi{100} \small (2,6) and \dpi{100} \small (3,5)?

Possible Answers:

\dpi{100} \small y=2x-4

\dpi{100} \small y=3x+2

\dpi{100} \small y=-x+8

\dpi{100} \small y=x+7

\dpi{100} \small y=2x+4

Correct answer:

\dpi{100} \small y=-x+8

Explanation:

Our first step will be to determing the slope of the line that connects the given points.

Our slope will be . Using slope-intercept form, our equation will be . Use one of the give points in this equation to solve for the y-intercept. We will use \dpi{100} \small (2,6).

Now that we know the y-intercept, we can plug it back into the slope-intercept formula with the slope that we found earlier.

This is our final answer.

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

Which of the following equations does NOT represent a line?

Possible Answers:

Correct answer:

Explanation:

The answer is .

A line can only be represented in the form  or , for appropriate constants , , and . A graph must have an equation that can be put into one of these forms to be a line.

 represents a parabola, not a line. Lines will never contain an term.

Example Question #12 : How To Find The Equation Of A Line

Let y = 3x – 6.

At what point does the line above intersect the following:

 

 

Possible Answers:

They do not intersect

(–5,6)

They intersect at all points

(–3,–3)

(0,–1)

Correct answer:

They intersect at all points

Explanation:

If we rearrange the second equation it is the same as the first equation. They are the same line.

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

Which of the following is the equation of a line between the points  and ?

Possible Answers:

Correct answer:

Explanation:

Since you have y-intercept, this is very easy.  You merely need to find the slope.  Then you can use the form  to find one version of the line.

The slope is:

Thus, for the points  and , it is:

Thus, one form of our line is:

If you move the  to the left side, you get:

, which is one of your options.

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

What is an equation of the line going through points  and ?

Possible Answers:

Correct answer:

Explanation:

If you have two points, you can always use the point-slope form of a line to find your equation.  Recall that this is:

You first need to find the slope, though.  Recall that this is:

For the points  and , it is:

Thus, you can write the equation using either point:

Now, notice that one of the options is:

This is merely a multiple of the equation we found, so it is fine!

 

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