Help with geometry please?

Question

1. parallel to 3x - 6y = 5, containing (-2, 4)

2. containing (-2, 6) and having slope -3

what are the steps to solve this problem/ equation?

1. y = x - 3

4x + y = 32

2. -7x + 8y = 32

5x + 6y = 24

Answer 1

1)  First solve the equation for y so that the equation will be in slope-intercept form, y = mx + b, where m is the slope of the  line and b is  the y-intercept (where the line crosses the y-axis):

3x - 6y = 5

-6y = -3x + 5

y = (1/2)x - (5/6)

The slope m = 1/2.  The y-intercept b = -5/6.

A line parallel to this line will have the same slope but a different y-intercept.

Since the parallel line has the same slope, m = 1/2.

So the equation for the parallel line so far is...

y = (1/2)x + b

Use the point given in the problem to solve for b.  Plug the point (-2,4) into x and y, and then solve for b:

y = (1/2)x + b

4 = (1/2)(-2) + b

4 = -1 + b

5 = b

So, the equation of the parallel line is y = (1/2)x + 5.

If you want the equation in standard form...

y = (1/2)x + 5

0 = (1/2)x - y + 5

-5 = (1/2)x - y

-10 = x - 2y

x - 2y = -10

2)  Use the method above.

You are given that the slope is -3. So, m = -3.

y = -3x + b

Plug the point (-2, 6) into the equation, and then solve for b:

y = -3x + b

6 = -3(-2) + b

6 = 6 + b

0 = b

So, the equation is y = -3x + 0, or y = -3x

--------------------------------------...

These are called systems of equations.  You can use substitution to solve these.

1)  y = x - 3; 4x + y = 32

You know that y = x - 3.  So, y is the same as x - 3.  Substitute x - 3 for y into the second equation:

4x + y = 32

4x + x - 3 = 32.  Solve for x:

5x - 3 = 32

5x = 35

x = 7

You now know that x equals 7.  Plug 7 into the first equation, and then you can find what y is:

y = x - 3

y = 7 - 3

y = 4

x = 7 & y = 4

2.  Since neither of the equations is solved for x or y, you have to solve one of them for x or y yourself.  Let's solve the first equation for x:

-7x + 8y = 32

-7x = -8y + 32

x = (8/7)y - (32/7)

You now know what x equals.  Plug that value into the second equation to find y:

5x + 6y = 24

5[(8/7)y - (32/7)] + 6y = 24

(40/7)y - (160/7) + 6y = 24

(82/7)y - (160/7) = 24

(82/7)y = 328/7

y = 4

Now that you have what y equals.  Plug that into the first equation to find what x equals:

-7x + 8y = 32

-7x + 8(4) = 32

-7x + 32 = 32

-7x = 0

x = 0

x = 0 & y = 4

Hope that helped!