Candace S. answered • 03/01/16

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A day without Math is like a day without sunshine!

Hi Jack,

1) Find the equation of the line:

a) Through the point (5-,8), parallel to the line 2x-7y=2

b) Though the point (4,0), perpendicular to the line 2x-7y=2

a) Through the point (5-,8), parallel to the line 2x-7y=2

b) Though the point (4,0), perpendicular to the line 2x-7y=2

Part a) Parallel lines have the same slope but different y-intercepts.

First we need to change the linear equation from standard form to slope-intercept form, y=mx+b.

2x-7y=2

add -2x to both sides: -7y=-2x+2

divide all terms by -7: y=2/7x -2/7

The slope is 2/7.

To write an equation of a parallel line through point (5,8) start with y=mx+b

substitute in the slope

y=2/7x+b

substitute in the point

8=2/7(5)+b

8=10/7 + b

Add -10/7 to both sides

46/7=b

The equation of the parallel line is y=2/7x+46/7

Part b) Though the point (4,0), perpendicular to the line 2x-7y=2

Perpendicular lines have slopes which are negative reciprocals.

As with part a we need to write the equation in slope intercept form

2x-7y=2

add -2x to both sides

-7y=-2x+2

divide all three terms by -7

y=2/7x-2/7

The slope is 2/7. So the negative reciprocal is -7/2.

y=mx+b

y=-7/2x+b

substitute in point (4,0)

0=-7/2(4)+b

0=-14+b

14=b

So the perpendicular line is y=-7/2x+14

2) Solve the following systems of equations for x,y,z:

a) 2x+y+z=6

3x+4y-z=4

5x-2y+z=36

In order to solve this system we must make sure to use all three equations. The solution is the point were all three planes intersect.

let's eliminate one variable first.

using 1st and 2nd equations, by add the together we can eliminate z-variable.

then using 2nd and 3rd equations we can do the same process

2x+y+z=6

3x+4y-z=4

3x+4y-z=4

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5x+5y=10

Something interesting about this equation is all the three coefficients are multiples of 5, so it can be simplified further by dividing each term by 5.

x+y=2

now let's focus on the 2nd and 3rd equations

3x+4y-z=4

5x-2y+z=36

5x-2y+z=36

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8x+2y=40

Something interesting about this equation is all the three coefficients are multiples of 2, so it can be simplified further by dividing each term by 2.

4x+y=20

Now we have two equations

4x+2y=20

x+y=2

we can eliminate a variable by subtracting the two equations

4x+y=20

- [x+y=2]

- [x+y=2]

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3x=18

x=6

substitute x=6 into the second equation to solve for y

x+y=2

6+y=2

y=-4

Now to find the last variable z, substitute in x=6 and y=-4 into one of the equations from our original problem.

2x+y+z=6

2(6)+-4+z=6

12+-4+z=6

8+z=6

z=-2

**ANSWER**: x=6, y=-4 and z=-2

Check:

3x+4y-z=4

3(6)+4(-4)-(-2)=4

18-16+2=4

2+2=4

4=4 it checks

We're finished

Jack N.

03/01/16