Other Lines - ACT Math

Card 0 of 20

Question

For the line

$y=\frac{1}{3}x-7$

Which one of these coordinates can be found on the line?

Answer

To test the coordinates, plug the x-coordinate into the line equation and solve for y.

y = 1/3x -7

Test (3,-6)

y = 1/3(3) – 7 = 1 – 7 = -6 YES!

Test (3,7)

y = 1/3(3) – 7 = 1 – 7 = -6 NO

Test (6,-12)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (6,5)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (9,5)

y = 1/3(9) – 7 = 3 – 7 = -4 NO

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Question

Consider the lines described by the following two equations:

4y = 3x2

3y = 4x2

Find the vertical distance between the two lines at the points where x = 6.

Answer

Since the vertical coordinates of each point are given by y, solve each equation for y and plug in 6 for x, as follows:

Taking the difference of the resulting y -values give the vertical distance between the points (6,27) and (6,48), which is 21.

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Question

Solve the following system of equations:

–2x + 3y = 10

2x + 5y = 6

Answer

Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)

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Question

Which of the following sets of coordinates are on the line $y=3x-4$ ?

Answer

$(2,2)$ when plugged in for $y$ and $x$ make the linear equation true, therefore those coordinates fall on that line.

$y=3x-4$

Because this equation is true, the point must lie on the line. The other given answer choices do not result in true equalities.

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Question

Which of the following points can be found on the line $\small y=3x+2$ ?

Answer

We are looking for an ordered pair that makes the given equation true. To solve, plug in the various answer choices to find the true equality.

Because this equality is true, we can conclude that the point lies on this line. None of the other given answer options will result in a true equality.

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Question

Which of the following points is on the line ?

Answer

The only thing that is necessary to solve this question is to see if a given value will provide you with the value paired with it. Among the options provided, only works. This is verified by the following simple substitution:

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Question

Given the graph of the line below, find the equation of the line.

Answer

To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

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Question

A line is defined by the following equation:

What is the slope of that line?

Answer

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

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Question

If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?

Answer

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

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Question

Which line passes through the points (0, 6) and (4, 0)?

Answer

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

First, calculate the slope: m = rise ÷ run = (y2 – y1)/(x2 – 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

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Question

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

Answer

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.

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Question

Let y = 3_x_ – 6.

At what point does the line above intersect the following:

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

Answer

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

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Question

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

Answer

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

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-5}{2-3}=\frac{1}{-1}=-1$

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.

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Question

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

Answer

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:

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

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

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

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Question

Which of the following equations does NOT represent a line?

Answer

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.

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Question

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

Answer

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:

$\frac{rise}{run} = \frac{y_1-y_0}{x_1-x_0}$

Thus, for the points and , it is:

$\frac{15-3}{4-0}= \frac{12}{4}=3$

Thus, one form of our line is:

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

, which is one of your options.

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Question

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

Answer

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:

$\frac{rise}{run} = \frac{y_1-y_0}{x_1-x_0}$

For the points and , it is:

$\frac{10-8}{6-4}=\frac{2}{2} = 1$

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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Question

What is the slope of the line:

$\frac{14}{3}x=\frac{1}{6}y-7$

Answer

First put the question in slope intercept form (y = mx + b):

–(1/6)y = –(14/3)x – 7 =>

y = 6(14/3)x – 7

y = 28x – 7.

The slope is 28.

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Question

If 2x – 4y = 10, what is the slope of the line?

Answer

First put the equation into slope-intercept form, solving for y: 2x – 4y = 10 → –4y = –2x + 10 → y = 1/2*x – 5/2. So the slope is 1/2.

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Question

What is the slope of the line with equation 4_x_ – 16_y_ = 24?

Answer

The equation of a line is:

y = mx + b, where m is the slope

4_x_ – 16_y_ = 24

–16_y_ = –4_x_ + 24

y = (–4_x_)/(–16) + 24/(–16)

y = (1/4)x – 1.5

Slope = 1/4

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