How to find the equation of a parallel line

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Math › How to find the equation of a parallel line

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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.

$\small y=x+1$

$\small y=-x+1$

$\small y=x+2$

$\small y=-x-2$

$\small y=2x+1$

Explanation

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 $\small (1,2)$ .

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

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

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

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

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

Explanation

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

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

A line is parallel to the line of the equation

and passes through the point .

Give the equation of the line in standard form.

None of the other choices gives the correct response.

Explanation

Two parallel lines have the same slope. Therefore, it is necessary to find the slope of the line of the equation

Rewrite the equation in slope-intercept form . , the coefficient of , will be the slope of the line.

Add to both sides:

Multiply both sides by $\frac{1}{7}$ , distributing on the right:

$\frac{1}{7} \cdot 7y = \frac{1}{7} \cdot( -5x+32)$

$y = -\frac{5}{7}x+\frac{32}{7}$

The slope of this line is $-\frac{5}{7}$ . The slope of the first line will be the same. The slope-intercept form of the equation of this line will be

$y =-\frac{5}{7}x+b$ .

To find , set and and solve:

$8 =-\frac{5}{7} (-2)+b$

$8 = \frac{10}{7} +b$

$\frac{56}{7} = \frac{10}{7} +b$

Subtract $\frac{10}{7}$ from both sides:

$\frac{56}{7} - \frac{10}{7} = \frac{10}{7} +b - \frac{10}{7}$

$b = \frac{46}{7}$

The slope-intercept form of the equation is $y =-\frac{5}{7}x+ \frac{46}{7}$

To rewrite in standard form with integer coefficients:

Multiply both sides by 7:

$7 y =7\left (-\frac{5}{7}x+ \frac{46}{7} \right )$

$7 y =7\left (-\frac{5}{7}x \right ) +7\left ( \frac{46}{7} \right )$

Add to both sides:

the correct equation in standard form.

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?

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

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

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

Explanation

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 $\small (0,3)$ .

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

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

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

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.

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

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

$y=-\frac{3}{4}x-10$

$y=\frac{4}{3}x-10$

$y=\frac{3}{4}x-10$

Explanation

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 $\small (4,13)$ .

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

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

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

Find the equation of a line parallel to the line that goes through points $\left ( 2,4 \right )$ and $\left ( 0,7 \right )$ .

$y=-\frac{3}{2}X+7$

$y=\frac{2}{3}X+7$

$y=\frac{3}{2}X-5$

Explanation

Parallel lines share the same slope. Because the slope of the original line is $-\frac{3}{2}$ , the correct answer must have that slope, so the correct answer is

$y=-\frac{3}{2}X+7$

Suppose a line . What is the equation of a parallel line that intersects point ?

Explanation

A line parallel to must have a slope of two. Given the point and the slope, use the slope-intercept formula to determine the -intercept by plugging in the values of the point and solving for :

Plug the slope and the -intercept into the slope-intercept formula:

Which one of these equations is parallel to:

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

$y=\frac{-1}{3}+4$

Explanation

Equations that are parallel have the same slope.

For the equation:

The slope is since that is how much changes with increment of .

The only other equation with a slope of is:

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

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

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

$y=\frac{2}{3}x-8$

Explanation

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.

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

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

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

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

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

Given the equation and the point , find a line through the point that is parallel to the given line.

$y=\frac{1}{2}x+8$

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

Explanation

In order for two lines to be parallel, they must have the same slope. The slope of the given line is , so we know that the line going through the given point also has to have a slope of . Using the point-slope formula,