Lines - PSAT Math

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Question

In the xy -plane, line l is given by the equation 2_x_ - 3_y_ = 5. If line l passes through the point (a ,1), what is the value of a ?

Answer

The equation of line l relates x -values and y -values that lie along the line. The question is asking for the x -value of a point on the line whose y -value is 1, so we are looking for the x -value on the line when the y-value is 1. In the equation of the line, plug 1 in for y and solve for x:

2_x_ - 3(1) = 5

2_x_ - 3 = 5

2_x_ = 8

x = 4. So the missing x-value on line l is 4.

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Question

The equation of a line is: 2x + 9y = 71

Which of these points is on that line?

Answer

Test the difference combinations out starting with the most repeated number. In this case, y = 7 appears most often in the answers. Plug in y=7 and solve for x. If the answer does not appear on the list, solve for the next most common coordinate.

2(x) + 9(7) = 71

2x + 63 = 71

2x = 8

x = 4

Therefore the answer is (4, 7)

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Question

$\dpi{100} \small 5x+25y = 125$

Which point lies on this line?

Answer

$\dpi{100} \small 5x+25y = 125$

Test the coordinates to find the ordered pair that makes the equation of the line true:

$\dpi{100} \small (5,4)$

$\dpi{100} \small 5 (5) + 25 (4) = 25 + 100 = 125$

$\dpi{100} \small (1,5)$

$\dpi{100} \small 5(1)+25(5)= 5+125=130$

$\dpi{100} \small (5,1)$

$\dpi{100} \small 5(5)+25(1)= 25+25=50$

$\dpi{100} \small (5,5)$

$\dpi{100} \small 5(5)+25(5)= 25+125=150$

$\dpi{100} \small (1,4)$

$\dpi{100} \small 5(1)+25(4)= 5+100=105$

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Question

Which of the following lines contains the point (8, 9)?

Answer

In order to find out which of these lines is correct, we simply plug in the values $\dpi{100} \small x=8$ and $\dpi{100} \small y=9$ into each equation and see if it balances.

The only one for which this will work is $\dpi{100} \small 3x-6=2y$

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Question

Points D and E lie on the same line and have the coordinates $\left ( 1,2\right )$ and $\left ( 5,7\right )$ , respectively. Which of the following points lies on the same line as points D and E?

Answer

The first step is to find the equation of the line that the original points, D and E, are on. You have two points, so you can figure out the slope of the line by plugging the points into the equation

$slope = \frac{\Delta y}{\Delta x} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ .

$slope = \frac{7-2}{5-1} = \frac{5}{4}$

Therefore, you can get an equation in the line in point-slope form, which is

.

Plug in the answer options, and you will find that only the point $\left ( 3,4.5\right )$ solves the equation.

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Question

Which of the following points is on the line given by the equation $y=\frac{1}{2}x+3$ ?

Answer

In order to solve this, try each of the answer choices in the equation:

$y=\frac{1}{2}x+3$

For example, when we try (3,4), we find:

$4=\frac{1}{2}(3)+3$

This does not work. When we try all the choices, we find that only (2,4) works:

$y=\frac{1}{2}x+3$

$4=\frac{1}{2}(2)+3$

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Question

Which line below is parallel to y – 2 = ¾x ?

Answer

y – 2 = ¾x is y = ¾x + 2 in slope intercept form (y=mx + b where m is the slope and b is the y-intercept). In this line, the slope is ¾. Parallel lines have the same slope.

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Question

All of the following systems of equations have exactly one point of intersection EXCEPT __________.

Answer

In order for two lines to intersect exactly once, they can't be parallel; thus, their slopes cannot be equal. If two lines have slopes that are indeed equal, these lines are parallel. Parallel lines either overlap infinitely or they never meet. If they overlap, they intersect at infinitely many points (which is not the same as intersecting exactly once).

In other words, we are looking for the system of equations with lines that are parallel, because then they will either intersect infinitely many times, or not at all. If the lines are not parallel, they will intersect exactly once.

The only system of equations that consists of parallel lines is the one that consists of the lines 4x - 3y = 2 and 6y = 8x + 9. To determine whether or not these lines are parallel, we need to find their slopes. It helps to remember that the slope of a line in the standard form Ax + By = C is equal to -A/B. (Alternatively, you can solve for the slopes by rearranging both lines to slope-intercept form).

The line 4x - 3y = 2 is already in standard form, so its slope is -4/-3 = 4/3.

The line 6y = 8x + 9 is not in standard form, so we must rearrange it a little bit. First let's subtract 6y from both sides.

0 = 8x - 6y + 9

Then we can subtract 9 from both sides.

8x - 6y = -9

Now that the equation is in standard form, the slope is -8/-6 = 4/3.

Thus, these two lines are parallel, so they will either intersect infinitely many times, or not at all.

If we check all of the other systems of equations, we will find that each consists of lines that aren't parallel. Thus, all the other choices consist of lines that intersect exactly once.

The answer is the system of lines 4x - 3y = 2 and 6y = 8x + 9.

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Question

Assume line a and line b are parallel.

If angle x is three bigger than twice the square of four of angle y, then what is angle y?

Answer

The answer is 7.

Line a and b are parallel lines cut by a transverse line which make angle x and y alternate exterior angles. This means that angle x and angle y have the same measurement value.

The square root of 4 is 2; so twice 2 is 4. Then three added to 4 is 7. So x is equal to 7 and thus y is also equal to 7.

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Question

Two lines are described by the equations:

y = 3x + 5 and 5y – 25 = 15x

Which of the following is true about the equations for these two lines?

Answer

The trick to questions like this is to get both equations into the slope-intercept form. That is done for our first equation (y = 3x + 5). However, for the second, some rearranging must be done:

5y – 25 = 15x; 5y = 15x + 25; y = 3x + 5

Note: Not only do these equations have the same slope (3), they are totally the same; therefore, they represent the same equation.

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Question

Line is given by the equation . All of the following lines intersect EXCEPT:

Answer

In order for two lines to intersect, they cannot be parallel. Thus, we need to look at each of the choices and determine whether or not each line is parallel to line q, given by the equation 2x – 3y = 4.

To see whether or not two lines are parallel, we must compare their slopes. Two lines are parallel if and only if their slopes are equal. The line 2x – 3y = 4 is in standard form. In general, a line in the form Ax + By = C has a slope of –A/B; therefore, the slope of line q must be –2/–3 = 2/3.

Let's look at the line 2x + 3y = 4. This line is also in standard form, so its slope is –2/3. Because the slope of this line is not equal to the slope of line q, the two lines aren't parallel. That means line 2x + 3y = 4 will intersect q at some point (we don't need to determine where).

Next, let's examine the line y = 4x – 5. This line is in slope-intercept form. In general, a line in the form y = mx + b has a slope equal to m. Thus, the slope of this line equals 4. Because the slope of this line is not the same as the slope of q, these lines will intersect somewhere. We can eliminate y = 4x – 5 from our answer choices.

Similarly, y = 3x is in slope-intercept form, so its slope is 3, which doesn't equal the slope of q. We can eliminate y = 3x from our choices.

Next, let's analyze 4x – 6y = 8. The slope of this line is –4/–6 = 2/3, which is equal to the slope of q. Thus, this line is parallel to q. However, just because two lines are parallel doesn't mean they will never intersect. If two lines overlap, they are parallel, and they will intersect infinitely many times. In order to determine if 4x – 6y = 8 intersects line q, let's find a point on q and see if this point is also on the line 4x – 6y = 8.

Line q has the equation 2x – 3y = 4. When x = –1, y = –2. This means that q passes through the point (–1, –2). Let's see if the line 4x – 6y = 8 also passes through the point (–1, –2) by substituting –1 and –2 in or x and y.

4(–1) –6 (–2) = –4 + 12 = 8

The line 4x – 6y = 8 also passes through the point (–1, –2). This means that this line overlaps with line q, and they intersect infinitely many times.

By process of elimination, we are left with the line –2x + 3y = 4. However, let's verify that these lines don't intersect. The slope of this line is –(–2)/3 = 2/3, so that means it is parallel to line q. Let's see if this line passes through the point (–1, –2).

–2(–1) + 3(–2) = 2 – 6 = –4, which doesn't equal 4. In other words, this line doesn't pass through the same point as q. This means that the line –2x + 3y = 4 is parallel to q, but the two lines don't overlap, and thus can never intersect.

The answer is –2x + 3y = 4.

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Question

A line passes through the points $(-1,-2)$ and $(1,2)$ . Which of the following lines is parallel to this line?

Answer

Lines are parallel if they have the same slope. First, let's find the slope of the line between $(-1,-2)$ and $(1,2)$ . $slope = \frac{rise}{run} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{2 + 2}{1 + 1} = \frac{4}{2} = 2$

So we are looking for a line with a slope of 2. We'll go through the answer choices.

The line between the points $(-2,0)$ and $(0,4)$ : $slope = \frac{4 - 0}{0 + 2} = 2$ . This is the same slope, so the lines are parallel, and this is the correct answer. We'll go through the rest of the answer choices for completeness.

$y=-3x+4$ : This is in the form $y=mx+b$ , where $m$ is the slope. Here the slope is $-3$ , so this is incorrect.

$y=\frac{x}{2}-4$ : Here the slope is $\frac{1}{2}$ , so this is again incorrect.

$y=-\frac{x}{2}+7$ : The slope is $-\frac{1}{2}$ , which is the negative reciprocal of 2. This line is perpendicular, not parallel, to the line in question.

The line between the points $(4,7)$ and $(7,4)$ : $slope = \frac{4 - 7}{7 - 4} = -1$ , also incorrect.

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Question

Which of the following lines is parallel with ?

Answer

Parallel lines have the same slope. Since the slope of is , we need to pick out the equation of another line that also has a slope of . Put each option in form can help you easily identify which line has a slope of :

becomes , which has a slope of .

$y=\frac{1}{2}x+4$ is already in form and has a slope of $\frac{1}{2}$ .

is also already in form and has a slope of .

$\frac{1}{3}x=y-6$ becomes $y=\frac{1}{3}x+6$ , which has a slope of $\frac{1}{3}$ .

$x=3+\frac{y}{3}$ becomes $\frac{y}{3}=x-3$ and then . This line has a slope of , so it is the correct answer.

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Question

Which set of lines is perpendicular?

Answer

Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.

y = 3_x_ + 5 and y = 5_x_ + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.

y = 3_x_/5 – 3 and y = 5_x_/3 + 3 both have positive slopes, so again they aren't perpendicular.

y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.

The line between the points (1,3) and (3,5), and y = 4_x_ + 7: We need to find the slope of the first line. slope = rise / run = (_y_2 – _y_1) / (_x_2 – x_1) = (5 – 3) / (3 – 1) = 1. The slope of y = 4_x + 7 is also positive (m = 4), so the lines are not perpendicular.

The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.

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Question

If two lines have slopes of $-5$ and $\frac{1}{5}$ , which statement about the lines is true?

Answer

Perpendicular lines have slopes that are the negative reciprocals of each other.

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Question

Which of the following lines is perpendicular to $y=3x-4$

Answer

The line which is perpendicular has a slope which is the negative inverse of the slope of the original line.

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Question

Which set of lines is perpendicular?

Answer

Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.

y = 3_x_ + 5 and y = 5_x_ + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.

y = 3_x_/5 – 3 and y = 5_x_/3 + 3 both have positive slopes, so again they aren't perpendicular.

y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.

The line between the points (1,3) and (3,5), and y = 4_x_ + 7: We need to find the slope of the first line. slope = rise / run = (_y_2 – _y_1) / (_x_2 – x_1) = (5 – 3) / (3 – 1) = 1. The slope of y = 4_x + 7 is also positive (m = 4), so the lines are not perpendicular.

The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.

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Question

Line includes the points and . Line includes the points and . Which of the following statements is true of these lines?

Answer

We calculate the slopes of the lines using the slope formula.

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{3- 9}{-20- (-12)} = \frac{-6}{-8} = \frac{3}{4}$

Parallel lines and identical lines must have the same slope, so these can be eliminated as choices. The slopes of perpendicular lines must have product . The slopes have product

$- \frac{3}{4} \times \frac{3}{4} = - \frac{9}{16} eq -1$

so they are not perpendicular.

The correct response is that the lines are distinct but neither parallel nor perpendicular.

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Question

Line includes the points and . Line includes the points and . Which of the following statements is true of these lines?

Answer

We calculate the slopes of the lines using the slope formula.

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-2 - 6}{5- 11} = \frac{-8}{-6} = \frac{4}{3}$

The slopes are not the same, so the lines are neither parallel nor identical. We multiply their slopes to test for perpendicularity:

$- \frac{3}{4} \cdot \frac{4} {3} = - \frac{12}{12} = -1$

The product of the slopes is , making the lines perpendicular.

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Question

Consider the equations and . Which of the following statements is true of the lines of these equations?

Answer

We find the slope of each line by putting each equation in slope-intercept form and examining the coefficient of .

is already in slope-intercept form; its slope is .

To get in slope-intercept form we solve for :

$\frac{- 2y }{-2}= \frac{- 8x + 18}{-2}$

The slope of this line is also .

The slopes are equal; however, the -intercepts are different - the -intercept of the first line is and that of the second line is . Therefore, the lines are parallel as opposed to being the same line.

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