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ACT Math : Other Lines

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

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.

Possible Answers:

Correct answer:

Explanation:

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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Example Question #481 : Geometry

For the line

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

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

Possible Answers:

(3, 7)

(9, 5)

(3, –6)

(6, –12)

(6, 5)

Correct answer:

(3, –6)

Explanation:

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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Example Question #4 : How To Find Out If A Point Is On A Line With An Equation

Solve the following system of equations:

–2x + 3y = 10

2x + 5y = 6

Possible Answers:

(3, –2)

(2, 2)

(3, 5)

(–2, 2)

(–2, –2)

Correct answer:

(–2, 2)

Explanation:

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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Example Question #3 : How To Find Out If A Point Is On A Line With An Equation

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

Possible Answers:

$(2,-2)$ $\displaystyle (2,-2)$

$(1,5)$ $\displaystyle (1,5)$

$(1,2)$ $\displaystyle (1,2)$

$(3,4)$ $\displaystyle (3,4)$

$(2,2)$ $\displaystyle (2,2)$

Correct answer:

$(2,2)$ $\displaystyle (2,2)$

Explanation:

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

$y=3x-4$ $\displaystyle y=3x-4$

$\displaystyle 2=3(2)-4$

2=6-4 $\displaystyle 2=6-4$

2=2 $\displaystyle 2=2$

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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Example Question #4 : How To Find Out If A Point Is On A Line With An Equation

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

Possible Answers:

(-1,2) $\displaystyle (-1,2)$

(1, 5) $\displaystyle (1, 5)$

(1, 0) $\displaystyle (1, 0)$

(2, 7) $\displaystyle (2, 7)$

(0, 1) $\displaystyle (0, 1)$

Correct answer:

(1, 5) $\displaystyle (1, 5)$

Explanation:

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.

y=3x+2 $\displaystyle y=3x+2$

$\displaystyle 5=3(1)+2$

5=3+2 $\displaystyle 5=3+2$

5=5 $\displaystyle 5=5$

Because this equality is true, we can conclude that the point (1,5) $\displaystyle (1,5)$ lies on this line. None of the other given answer options will result in a true equality.

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Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Which of the following points is on the line $\displaystyle y=15x + 22$ ?

Possible Answers:

$\displaystyle (1.5,32)$

$\displaystyle (6.5,80.5)$

(4,104) $\displaystyle (4,104)$

$\displaystyle (2.5,66)$

$\displaystyle (5.5,104.5)$

Correct answer:

$\displaystyle (5.5,104.5)$

Explanation:

The only thing that is necessary to solve this question is to see if a given $\displaystyle x$ value will provide you with the $\displaystyle y$ value paired with it. Among the options provided, only $\displaystyle (5.5,104.5)$ works. This is verified by the following simple substitution:

$\displaystyle y = 15 * 5.5 + 22$

y=104.5 $\displaystyle y=104.5$

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Example Question #83 : Coordinate Plane

What is the slope of line 3 = 8y - 4x?

Possible Answers:

-2

-0.5

0.5

Correct answer:

0.5

Explanation:

Solve equation for y. y=mx+b, where m is the slope

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Example Question #84 : Coordinate Plane

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

Possible Answers:

–0.5

0.5

–5/2

–2

Correct answer:

0.5

Explanation:

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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Example Question #1411 : Gre Quantitative Reasoning

What is the slope of the line with equation 4x – 16y = 24?

Possible Answers:

–1/4

1/4

1/2

–1/8

1/8

Correct answer:

1/4

Explanation:

The equation of a line is:

y = mx + b, where m is the slope

4x – 16y = 24

–16y = –4x + 24

y = (–4x)/(–16) + 24/(–16)

y = (1/4)x – 1.5

Slope = 1/4

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Example Question #31 : Lines

What is the slope of a line which passes through coordinates $\dpi{100} \small (3,7)$ $\displaystyle \dpi{100} \small (3,7)$ and $\dpi{100} \small (4,12)$ $\displaystyle \dpi{100} \small (4,12)$ ?

Possible Answers:

$\dpi{100} \small \frac{1}{5}$

$\dpi{100} \small 5$ $\displaystyle \dpi{100} \small 5$

$\dpi{100} \small \frac{1}{2}$

$\dpi{100} \small 3$ $\displaystyle \dpi{100} \small 3$

$\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$

Correct answer:

$\dpi{100} \small 5$ $\displaystyle \dpi{100} \small 5$

Explanation:

Slope is found by dividing the difference in the $\dpi{100} \small y$ $\displaystyle \dpi{100} \small y$ -coordinates by the difference in the $\dpi{100} \small x$ $\displaystyle \dpi{100} \small x$ -coordinates.

$\dpi{100} \small \frac{(12-7)}{(4-3)}=\frac{5}{1}=5$

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ACT Math : Other Lines

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Example Question #481 : Geometry

Example Question #4 : How To Find Out If A Point Is On A Line With An Equation

Example Question #3 : How To Find Out If A Point Is On A Line With An Equation

Example Question #4 : How To Find Out If A Point Is On A Line With An Equation

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Example Question #83 : Coordinate Plane

Example Question #84 : Coordinate Plane

Example Question #1411 : Gre Quantitative Reasoning

Example Question #31 : Lines

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