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

To draw a line you either need twoof its points, or one of its points and its slope. Let's use this second approach.

We already have the point #(4,6)#. We derive the slope from the parallel line.

First of all, two lines are parallel if and only if they have the same slope. So, our line will have the same slope as the given line.

Secondly, to derive the slope from a line, we write its equation in the #y=mx+q# form. The slope will be the number #m#.

In this case, the line is already in this form, so we immediately see that the slope is #1/4#.

Recapping: we need a line passing through #(4,6)# and having slope #1/4#. The formula that gives the line equation is the following:

#y-y_0 = m(x-x_0)#

where #(x_0,y_0)# is the known point, and #m# is the slope. Let's plug our values:

#y-6= 1/4(x-4)#

Expanding the right hand side:

#y-6 = 1/4x-1#

Add #6# to both sides:

#y= 1/4x-1+6#

So the answer is

#y= 1/4x+5#

Parallel lines are coplanar lines that do not intersect. In two dimensions, parallel lines have the same slope .

We can write the equation of a line parallel to a given line if we know a point on the line and an equation of the given line.

Example:

Write the equation of a line that passes through the point ( 3 , 1 ) and is parallel to the line

y = 2 x + 3 .

Parallel lines have the same slope.

The slope of the line with equation y = 2 x + 3 is 2 . So, any line parallel to y = 2 x + 3 has the same slope 2 .

Now use the point-slope form to find the equation.

y − y 1 = m ( x − x 1 )

We have to find the equation of the line which has slope 2 and passes through the point ( 3 , 1 ) . So, replace m with 2 , x 1 with 3 , and y 1 with 1 .

y − 1 = 2 ( x − 3 )

Use the distributive property .

y − 1 = 2 x − 6

Add 1 to each side.

y − 1 + 1 = 2 x − 6 + 1 y = 2 x − 5

Therefore, the line y = 2 x − 5 is parallel to the line y = 2 x + 3 and passes through the point ( 3 , 1 ) .

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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?

Correct answer:

Explanation:

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

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

Correct answer:

Explanation:

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.

The slope of the line will be . In slope intercept-form, we know that the line will be . Now we can use the given point to find the y-intercept.

The final equation for the line will be .

What line is parallel to   and passes through the point ?

Correct answer:

Explanation:

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

The slope of this line is . 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.

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

What is the equation of a line that is parallel to the line and includes the point ?

Correct answer:

Explanation:

The line parallel to must have a slope of , giving us the equation . To solve for b, we can substitute the values for y and x.

Therefore, the equation of the line is .

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

Correct answer:

Explanation:

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

For parallel lines, the slopes must be equal, so the slope of the new line must also be . 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.

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

What line is parallel to  at ?

Possible Answers:

None of the answers are correct

Correct answer:

Explanation:

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

 therefore the slope is 

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

So, 

Thus, the new equation is 

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

Correct answer:

Explanation:

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 .

So the slope of the original line is . 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 . The second step is finding which line will give you that slope. For the correct answer, we find the following:

So, the slope is , and this line is perpendicular to the original.

Which of the following is a line that is parallel to the line defined by the equation ?

Correct answer:

Explanation:

Since parallel lines have equal slopes, you should find the slope of the line given to you. The easiest way to do this is to solve the equation so that its form is .   represents the slope.

Take your equation: 

First, subract  from both sides:

Next, subtract  from both sides:

Finally, divide by :

, which is the same as 

Thus, your slope is .

Among the options provided only  is parallel. Solve this equation as well for  form.  

First, subtract  from both sides:

Then, divide by :

Which of the following answer choices gives the equation of a line parallel to the line:

Correct answer:

Explanation:

Parallel lines have the same slope but different y-intercepts. When the equations of two lines are the same they have infinitely many points in common, whereas parallel lines have no points in common.

Our equation is given in slope-intercept form,

where  is the slope. In this particular situation .

Therefore we want to find an equation that has the same  value and a different  value.

Thus,

 is parallel to our equation.

What is the equation of a line parallel to the line given by the equation:
?

Correct answer:

Explanation:

Parallel lines have the same slope and differing y-intercepts. Since  is the only equation with the same slope, and the y-intercept is different, this is the equation of the parallel line.

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