Algebra - How to find the equation of a parallel line

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 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 $\small (1,2)$ .

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

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 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.

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

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

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

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

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

Which of the lines is parallel to ?

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

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

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

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

Explanation

In order for the lines to be parallel, both lines must have similar slope.

The current linear equation is in standard form. Rewrite this equation in slope intercept form, .

The slope is represented by the in the equation.

Subtract on both sides.

Simplify the left side and rearrange the right side.

Divide by nine on both sides.

$\frac{9y}{9}=\frac{-3x+5}{9}$

Simplify both sides of the equation.

$y=-\frac{1}{3}x+\frac{5}{9}$

The slope of this line is $-\frac{1}{3}$ .

The only line provided that has the similar slope is: $y=-\frac{1}{3}x+2$

The answer is: $y=-\frac{1}{3}x+2$

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

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

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

$2y=\frac{3}{4}+9$

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

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

Explanation

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

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

$y=-\frac{1}{2}x+6$

$y=\frac{1}{2}x+100$

$y=-\frac{2}{3}x-12$

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

Explanation

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: $y=-\frac{1}{2}x+6$

Find the line that is parallel to

and contains the point .

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

Explanation

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 .

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

None of the other answers.

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

$y=-\frac{1}{2}x-5$

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

A line cannot pass through this point and be parallel to the original line.

Explanation

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 .