How to find the equation of a parallel line - Algebra 1

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Question

Find a line parallel to

Answer

A parallel line will have the same value, in this case , as the orignal line but will intercept the at a different location.

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Question

Which of the following lines are parallel to the line defined by the equation:

Answer

Parallel means the same slope:

Solve for :

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

$m=\frac{1}{3}$

Find the linear equation where

$m=\frac{1}{3}$ .

$y=\frac{x}{3}+72$

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Question

What is the equation of the line parallel to that passes through (1,1)?

Answer

The line parallel to will have the same slope.

The equation for our parallel line will be: $\ y=4x+b$ .

Using the point (1,1) we can solve for the y-intercept:

$\ 1=4+b$

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Question

Which of these lines is parallel to

Answer

Parallel lines have identical slopes. To determine the slope of the given line, convert it to form:

2y = 3x + 8

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

This line has a slope of $\frac{3}{2}$ .

The only answer choice with a slope of $\frac{3}{2}$ is $y=\frac{3}{2}x-6$ .

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Question

Which of these lines is parallel to ?

Answer

Parallel lines have identical slopes. If you convert the given equation to the form , it becomes

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

The slope of this equation is $-\frac{1}{2}$ , so its parallel line must also have a slope of $-\frac{1}{2}$ . The only other line with a slope of $-\frac{1}{2}$ is

$y=-\frac{1}{2}x-4$

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Question

Choose which of the four equations listed is parallel to the given equation.

$y=\frac{3}{4}x+7$

Answer

$2y=\frac{3}{2}x+3$ is the correct answer because when each term is divided by 2 in order to see the equation in terms of y, the slope of the equation is $\frac{3}{4}$ , which is the same as the slope in the given equation. Parallel lines have the same slope.

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Question

Write an equation for a line that is parallel to $y=-\frac{1}2{}x+6$ and has a y-intercept of .

Answer

The equation of a line can be written using the expression where is the slope and is the y-intercept. When lines are parallel to each other, it means that they have the same slope, so $m=-\frac{1}{2}$ . The y-intercept is given in the problem as . This means that the equation would be $y=-\frac{1}{2}x-5$ .

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Question

Write the equation for a line parallel to $\small y=\frac{1}{4}x-5$ passing through the point $\small (4, -2)$ .

Answer

In order to approach this problem, we need to be familiar with the slope-intercept equation of a line, $\small y=mx+b$ where m is the slope and b is the y-intercept. The line that our line is supposed to be parallel to is $\small y=\frac{1}{4}x-5$ . Lines that are parallel have the same slope, m, so the slope of our new line is $\small \frac{1}{4}$ . Since we don't know the y-intercept yet, for now we'll write our equation as just:

$\small y=\frac{1}{4}x+b$ . We can solve for b using the point we know the line passes though, $\small (4, -2)$ . We can plug in 4 for x and -2 for y to solve for b:

$\small -2 = \frac{1}{4}4 + b$ first we'll multiply $\small \frac{1}{4}4$ to get 1:

$\small \small -2 = 1 + b$ now we can subtract 1 from both sides to solve for b:

$\small -3 = b$

Now we can just go back to our equation and sub in -3 for b:

$\small y = \frac{1}{4}x -3$

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Question

Find the equation for the line that goes through the point and is parallel to the line with the equation $y=\frac{2}{3}x+4$ .

Answer

The first thing we need to do is determine the slope of this parallel line. Recall that parallel lines have the same slope, so the slope of this new line must be $\frac{2}{3}$ .

Next, we must determine the -intercept of the line. The equation for a line is given to us by , and here we know three of the four variables in this new line: $m=\frac{2}{3},x=3,y=8$ .

So when we plug in these numbers we get

$8=\frac{2}{3}(3)+b$

Multiply to get:

Subtract from both sides to get:

Therefore, the equation for the line is:

$y=\frac{2}{3}x+6$

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Question

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

Answer

Parallel lines have the same slope. So our line should have a slope of 2x. Next we use the point slope formula to find the equation of the line that passes through and is parallel to .

Point slope formula:

$y-y_{1}=m(x-x_{1})$

is the slope of the line parallel to which passes through .

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Question

Find the equation of the line parallel to the given criteria: $m=-\frac{3}{2}$ and that passes through the point

Answer

Parallel lines have the same slope, so the slope of the new line will also have a slope $m=-\frac{3}{2}$

Use point-slope form to find the equation of the new line.

$y-y_{1}=m(x-x_{1})$

Plug in known values and solve.

$y-(-5)=-\frac{3}{2}(x-2)$

$y+5=-\frac{3}{2}x-\frac{-(3)}2{}2$

$y+5=-\frac{3}{2}x+3$

$y=-\frac{3}{2}x+3-5$

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

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Question

A line parallel to $\small y=x+6$ passes through the points $\small (1,2)$ and $\small (4,5)$ . Find the equation of this line.

Answer

This problem can be easily solved through using the point-slope formula:

where $\small m$ is the slope and $\small x_1$ and $\small y_1$ signify one of the given points (coordinates).

The problem provides us with two points, so that requirement is fulfilled. We may choose either one when substituting in our values. The only other requirement left is slope. The problem also provides us with this information, but it's not as obviously given. The problem specifies that the line of interest is parallel to $\small y=x+6$ . By definition, lines are parallel when they have the same slope. Given that information, if the two lines are parallel, the line of interest will have the same slope as the given equation: $\small m=1$ . Therefore, we have our required point and the slope. Now we may substitute in all the information and solve for the equation.

Here, we arbitrarily choose the point $\small (1,2)$ .

${\color{Blue} y=x+1}$

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Question

A line that passes through the points and is parallel to a line that has a slope of $\frac{1}{2}$ . What is the equation of this line?

Answer

This problem can be easily solved through using the point-slope formula:

where $\small m$ is the slope and $\small x_1$ and $\small y_1$ signify one of the given points (coordinates).

The problem provides us with two points, so that requirement is fulfilled. We may choose either one when substituting in our values. The only other requirement left is slope. The problem also provides us with this information, but it's not as obviously given. The problem specifies that the line of interest is parallel to a line with a slope of $\small \frac{1}{2}$ . By definition, lines are parallel when they have the same slope. Given that information, if the two lines are parallel, the line of interest will have the same slope as the given equation: $\small m=\frac{1}{2}$ . Therefore, we have our required point and the slope. Now we may substitute in all the information and solve for the equation.

Here, we arbitrarily choose the point $\small (0,3)$ .

$y-3=\frac{1}{2}(x-0)$

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

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

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Question

A line parallel to $y=\frac{3}{4}x+2$ passes through the points $\small (1, \frac{43}{10})$ and $\small (4,13)$ . Find the equation for this line.

Answer

This problem can be easily solved through using the point-slope formula:

where $\small m$ is the slope and $\small x_1$ and $\small y_1$ signify one of the given points (coordinates).

The problem provides us with two points, so that requirement is fulfilled. We may choose either one when substituting in our values. The only other requirement left is slope. The problem also provides us with this information, but it's not as obviously given. The problem specifies that the line of interest is parallel to $y=\frac{3}{4}x+2$ . By definition, lines are parallel when they have the same slope. Given that information, if the two lines are parallel, the line of interest will have the same slope as the given equation: $\small m=\frac{3}{4}$ . Therefore, we have our required point and the slope. Now we may substitute in all the information and solve for the equation.

Here, we arbitrarily choose the point $\small (4,13)$ .

$y-13=\frac{3}{4}(x-4)$

$y-13=\frac{3}{4}x-3$

${\color{Blue} y=\frac{3}{4}x+10}$

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Question

If is parallel to and passes through the point , what is the equation of ?

Answer

If f(x) is parallel to g(x) and passes through the point (-4,8), what is the equation of f(x)?

If f(x) is parallel to g(x), then it must have the same slope.

So,

$m_{f(x)}=5$

We also know that f(x) goes through the point (-4,8)... Set up the following:

Where b is our y-intercept, which we need to solve for:

Finally, put it all together:

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Question

Find the line that is parallel to

and contains the point .

Answer

When finding a line that is parallel to another line, we know that the slopes must be the same. So in the equation,

we know it has a slope of 2. We also know the parallel line contains the point

(-1, 5)

So, we will substitute the slope as well as the point into the y-intercept formula:

Doing this, we will find b, or the y-intercept, and we can determine the line that is parallel.

$5 = 2 \cdot -1 + b$

Now, we know the slope of the parallel line is still 2, and now we know the y-intercept is 7. Knowing this, we get the line

.

Therefore, is parallel to .

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Question

Which of the following lines is parallel to the following line:

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

Answer

Parallel lines have the same slope and the only equation that has the same slope as the given equation is

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

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Question

Which of the following equations will be parallel to the line connected to the points and ?

Answer

In order to determine the equation, we will need to find the slope of the line connected by the two given points.

Use the slope formula to determine the slope.

$m=\frac{y_2-y_1}{x_2-x_1}$

Substitute the points.

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

Our equation parallel this line connected by the two points must have a slope of negative one-half.

The only answer that has that slope is: $y=-\frac{1}{2}x+6$

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Question

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

Answer

When finding the slope of a parallel line, we need to ensure we have form. stands for slope. Our is which is also the slope of the parallel line. Since we are looking for an equation, we need to reuse the form to solve for . We do this by plugging in our coordinates.

Subtract on both sides.

Our equation is .

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Question

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

Answer

When finding the slope of a parallel line, we need to ensure we have form. By subtracting on both sides and dividing on both sides, we get

$y=\frac{3-10x}{12}$ .

stands for slope. Our is $-\frac{10}{12}$ or $-\frac{5}{6}$ which is also the slope of the parallel line. Since we are looking for an equation, we need to reuse the form to solve for . We do this by plugging in our coordinates.

$8=-\frac{5}{6}*-6+b$

Subtract on both sides.

Our equation is

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

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