# Math Help - Parallel and Perpendicular lines

1. ## Parallel and Perpendicular lines

Write an equation in slope-intercept form of the line in parallel to the graph of each equation and passes through the given point.

1. y=4x+5; (2,-3)

2. y=-1/2x-3; (0,0)

3. y=4; (2, 5)

4. 3x+y=3; (3, 5)

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Write an equation in slope-intercept form of the line that is perpendicular to the graph of each equation and passes through the given point.

1. y=2x+3; (3, -4)

2. y=-4x+5; (1, 1)

3. 2x-5y=3; (-2,7)

4. x=6; (4,2)
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Determine whether AB or CD are parallel, perpendicular, or neither.

1. A(4, 3), B(5, 2), C(8, 3), D(9,2)
2. A(-2, 3), B(4,0), C(2, 5), D(-1, 11)
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Wirte an equation in slop intercept form of each line.

• passes through the point at (4, -2) and is parallel to the graph of 5x-2y=6
• perpendicular to the line through points at (1,2) and (8,6) and passes through the point at (0, 5)

2. Consider the slope of each line, for a parallel line they will have the same slope. Parallel line have the same slope because they have to change at the same rate or else they would cross each other eventually right?

Perpendicular lines have slopes that are negative reciprocals of eachother i.e.

the negative reciprocal of $\frac{1}{2} is \frac{-2}{1}$ If we multiplied these together we would get -1 that would be the case with all perpendicular lines, if you multiply the slopes together you should get -1

3. So since we are worried about slopes putting them in the standard slope intercept form would be a good place to start. y=mx+b

4. Originally Posted by Temperamental
Write an equation in slope-intercept form of the line in parallel to the graph of each equation and passes through the given point. Note the parallel part.

1. y=4x+5; (2,-3)

2. y=-1/2x-3; (0,0)

3. y=4; (2, 5)

4. 3x+y=3; (3, 5)
Because it's parallel, the equation of the second line will have the same slope.

1) $y_2 = 4x + c$

Substitute $x = 2 ; y = -3$ (Given point)

$(-3) = (4)(2) + c$

Solve for c.

$c = -11$

Thus: $y_2 = 4x - 11$

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Same story. Gradient is the same, but we don't know $c$.

2) $y_2 = - \frac{1}{2} x + c$

Substitute $x = 0 ; y = 0$ (Given point)

$0 = 0 + c$

Solve for c.

$c = 0$

Thus: $y_2 = - \frac{1}{2} x + c$

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3) $y=4$

Try to envision this line. It's a horizontal line.

Basically any $y = c$ line would be parallel to this, but they specifically ask for the one which goes through the point where y = 5.

So $y_2 = 5$

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Try number 4 for yourself.
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Originally Posted by Temperamental
Write an equation in slope-intercept form of the line that is perpendicular to the graph of each equation and passes through the given point.

1. y=2x+3; (3, -4)

2. y=-4x+5; (1, 1)

3. 2x-5y=3; (-2,7)

4. x=6; (4,2)
Same as the first ones we did. Except here the gradients are not equal, but their products equal -1. Always. If the product of lines' gradients equal -1 then they are perpendicular.

I'll do number 1.

1.) $y=2x+3$

Now the gradient of $y_2$ should be the negative reciprocal of $y$.

$m_{y_{2}} = - \frac{1}{2}$

$y_2 = - \frac{1}{2} x + c$

You know the drill, substitute $(3, -4)$ and solve for $c$.
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Originally Posted by Temperamental
Determine whether AB or CD are parallel, perpendicular, or neither.

1. A(4, 3), B(5, 2), C(8, 3), D(9,2)
2. A(-2, 3), B(4,0), C(2, 5), D(-1, 11)
You know now that when gradients are equal they're parallel and when their product is -1 they're perpendicular. They could also be neither.

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Originally Posted by Temperamental
Wirte an equation in slop intercept form of each line.

• passes through the point at (4, -2) and is parallel to the graph of 5x-2y=6
• perpendicular to the line through points at (1,2) and (8,6) and passes through the point at (0, 5)
Same drill as all the other stuff.

5. Originally Posted by janvdl
Because it's parallel, the equation of the second line will have the same slope.

1) $y_2 = 4x + c$

Substitute $x = 2 ; y = -3$ (Given point)

$(-3) = (4)(2) + c$

Solve for c.

$c = -11$

Thus: $y_2 = 4x - 11$
What happened to the 5? why did you replace it with c?

6. Originally Posted by janvdl
...
2) $y_2 = - \frac{1}{2} x + c$

Substitute $x = 0 ; y = 0$ (Given point)

$0 = 0 + c$

Solve for c.

$c = 0$

Thus: $\boxed{\color{blue}y_2 = - \frac{1}{2} x }$
Originally Posted by Temperamental
What happened to the 5? why did you replace it with c?
The equation of the given line is: $y = 4x+5$

A parallel to this line has the same slope (here m = 4) but a different constant summand. This summand is unknown to us and therefore janvdl labeled it c. And you were asked to calculate the value of c using the fact that the parallel has to pass through the point P(2, -3), that means the coordinates of the point must satisfy the equation of the parallel:

To satisfy an equation means: If you plug in the x- and y-values the equation must be true. And that's the reason why you can use it to calculate the value of c.