How to find the equation of a parallel line - SSAT Upper Level Quantitative

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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

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 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 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 gives a line that is parallel to the line with the equation ?

Answer

Two lines are parallel when they have the same slope. Because the slope of the given line is , the slope to a line parallel to it must also be . The only answer choice that has a slope of is , so it is the correct answer.

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Question

A line has the equation . If a second line goes through the point and is parallel to the first line, what is the equation of this second line?

Answer

The slope of the second line must be if these two lines are to be parallel.

To find the equation of this second line, just plug in the given point into the standard form equation to find its -intercept.

Now we have all the parts needed to write the equation for the second line:

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Question

Line is parallel to line and goes through the point . The equation for line is . Find the equation of line .

Answer

Since lines and are parallel, the slope of line must also be . Now, plug the given point into the equation to find the -intercept of line :

Thus, the equation of line is

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Question

Find the equation of a line that goes through the point and is parallel to the line with the equation .

Answer

For lines to be parallel, they must have the same slope. The slope of the line we are looking for then must be .

The point that's given in the equation is also the y-intercept.

Using these two pieces of information, we know that the equation for the line must be

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Question

Find the equation of the line that goes through the point and is parallel to the line with the equation .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be .

We can then plug in the given point and the slope into the equation of a line to find the y-intercept.

Now, we can write the equation of the line.

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Question

Find the equation of the line that passes through the point and is parallel to the line with the equation $y=\frac{3}{2}x-1$ .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be $\frac{3}{2}$ .

Now, we can plug in the point given by the question to find the y-intercept.

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

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

$b=-\frac{7}{2}$

From this, we can write the following equation:

$y=\frac{3}{2}x-\frac{7}{2}$

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Question

Find the equation of the line that passes through the point and is parallel to the line with the equation .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be .

Next, plug in the point given by the question to find the y-intercept of the line.

Now, we know that the equation of the line must be .

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Question

Find the equation of the line that passes through the point and is parallel to the line with the equation .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be .

Next, plug in the point given by the question to find the y-intercept of the line.

Now, we know the equation of the line must be .

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Question

Find the equation of the line that passes through the point and is parallel to the line with the equation .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be .

Next, plug in the point given by the question to find the y-intercept of the line.

Now, we can write the equation for the line:

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Question

Find the equation of the line that passes through the point and is parallel to the line with the equation $y=\frac{1}{2}x-2$ .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be $\frac{1}{2}$ .

Next, plug in the point given by the question to find the y-intercept of the line.

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

Now, we knwo the equation of the line must be $y=\frac{1}{2}x+4$ .

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Question

Find the equation of the line that passes through the point $\left(\frac{1}{2}, 4\right)$ and is parallel to the line with the equation .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be .

Next, plug in the point given by the question to find the y-intercept of the line.

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

Now, we know the equation of the line must be .

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Question

Find the equation of the line that passes through the point $\left(2, \frac{1}{4}\right)$ and is parallel to the line with the equation $y=\frac{1}{8}x-2$ .

Answer

Because the two lines are parallel, we know that the slope of the line we need to find must also be $\frac{1}{8}$ .

Next, plug in the point given by the question to find the y-intercept of the line.

$\frac{1}{4}=\frac{1}{8}(2)+b$

$b+\frac{1}{4}=\frac{1}{4}$

.

We can then write the equation of the line: $y=\frac{1}{8}x$

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