Parallel Lines - SAT Math

Card 0 of 296

Question

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 ?

Answer

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.

$\frac{p--3}{-2-5}=\frac{p+3}{-2-5}=-2$

$\frac{p+3}{-7}=-2$

$(\frac{p+3}{-7})-7=-2(-7)$

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Question

Which of the following is the equation of a line that is parallel to the line 4_x_ – y = 22 and passes through the origin?

Answer

We start by rearranging the equation into the form y = mx + b (where m is the slope and b is the y intercept); y = 4_x_ – 22
Now we know the slope is 4 and so the equation we are looking for must have the m = 4 because the lines are parallel. We are also told that the equation must pass through the origin; this means that b = 0.

In 4_x_ – y = 0 we can rearrange to get y = 4_x_. This fulfills both requirements.

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Question

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?

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

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Question

What line is parallel to at ?

Answer

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

$y=\frac{-2}{3}x+2$ therefore the slope is $\frac{-2}{3}$

Parallel lines have the same slope, so now we need to find the equation of a line with slope $\frac{-2}{3}$ and going through point by substituting values into the point-slope formula.

$2=\frac{-2}{3}(3)+b$

So,

Thus, the new equation is $y=\frac{-2}{3}x+4$

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Question

What line is parallel to 2x + 5y = 6 through (5, 3)?

Answer

The given equation is in standard form and needs to be converted to slope-intercept form which gives y = –2/5x + 6/5. The parallel line will have a slope of –2/5 (the same slope as the old line). The slope and the given point are substituted back into the slope-intercept form to yield y = –2/5x +5.

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Question

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

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.

$y=-\frac{3}{5}x+\frac{8}{5}$

The slope of the line will be $-\frac{3}{5}$ . In slope intercept-form, we know that the line will be $y=-\frac{3}{5}x+b$ . Now we can use the given point to find the y-intercept.

$y=-\frac{3}{5}x+b$

$4=-\frac{3}{5}(10)+b$

The final equation for the line will be $y=-\frac{3}{5}x+10$ .

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Question

What line is parallel to and passes through the point ?

Answer

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

$y=-\frac{3}{2}+3$

The slope of this line is $-\frac{3}{2}$ . 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.

$y=-\frac{3}{2}x+b$

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

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

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

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Question

What line is parallel to through ?

Answer

The slope of the given line is and a parallel line would have the same slope, so we need to find a line through with a slope of 2 by using the slope-intercept form of the equation for a line. The resulting line is which needs to be converted to the standard form to get .

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Question

What is the equation of a line that is parallel to the line $\small y=\frac{1}{2}x+3$ and includes the point ?

Answer

The line parallel to $\small y=\frac{1}{2}x+3$ must have a slope of $\frac{1}{2}$ , giving us the equation $\small y=\frac{1}{2}x+b$ . To solve for b, we can substitute the values for y and x.

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

$\small 2=2+b$

$\small b=0$

Therefore, the equation of the line is $\small y=\frac{1}{2}x$ .

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Question

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

Answer

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

$y = \frac{-5}{3}x + \frac{8}{3}$

For parallel lines, the slopes must be equal, so the slope of the new line must also be $\frac{-5}{3}$ . 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.

$-4 = \frac{-5}{3}(6) + b$

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

$y = \frac{-5}{3}x + 6$

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Question

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

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 .

$y = \left(\frac{-2}{3}\right)x +\left(\frac{16}{3}\right)$

So the slope of the original line is $-\frac{2}{3}$ . 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 $\frac{3}{2}$ . The second step is finding which line will give you that slope. For the correct answer, we find the following:

$y= \left(\frac{3}{2}x\right) +\left(\frac{16.5}{6}\right)$

So, the slope is $\frac{3}{2}$ , and this line is perpendicular to the original.

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Question

Which of the following equations is parallel to: $y=\frac {2}{3}x+4$ and goes through the point ?

Answer

Step 1: We need to define what a parallel line is. A parallel line has the same slope as the line given in the problem. Parallel lines never intersect, which tells us that the y-intercepts of the two equations are different.

Step 2: We need to identify the slope of the line given to us. The slope is always located in front of the .

The slope in the equation is $\frac {2}{3}$ .

Step 3: If we said that a parallel line has the same slope as the given line in the equation, the slope of the parallel equation is also $\frac {2}{3}$ .

Step 4. We need to write the equation of the parallel line in slope-intercept form:. We need to write b for the intercept because it has changed.

The equation is: $y=\frac {2}{3}x+b$

Step 5: We will use the point where and . We need to substitute these values of x and y into the equation in step 4 and find the value of b.

$y=\frac {2}{3}x+b$
$5=\frac {2}{3}3+b$
$5=\frac {2}{{\color{Red} 3}}{\color{Red} 3}+b$
The numbers in red will cancel out when I multiply.

To find b, subtract 2 to the other side

Step 6: We put all of the parts together and make the final equation of the parallel line:

The final equation is: $y=\frac {2}{3}x+3$

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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

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 to:

Answer

First write the equation in slope intercept form. Add to both sides to get . Now divide both sides by to get . The slope of this line is , so any line that also has a slope of would be parallel to it. The correct answer is $x - \frac{1}{3}y = 7$ .

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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

Which pair of linear equations represent parallel lines?

Answer

Parallel lines will always have equal slopes. The slope can be found quickly by observing the equation in slope-intercept form and seeing which number falls in the " $m$ " spot in the linear equation $(y=mx+b)$ ,

We are looking for an answer choice in which both equations have the same $m$ value. Both lines in the correct answer have a slope of 2, therefore they are parallel.

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