x-2y+3z=7,2x+y+z=4,-3x+2y-2z=-10 i would like help to solve by using substitution and elimination

Question

x-2y+3z=7,2x+y+z=4,-3x+2y-2z=-10 i would like help to solve by using substitution and elimination

Answer 1

Hi Joan

 

We have 3 equations, which I'm going to number for easier reference.

 

1:    x   -2y +3z =7

2:   2x   +y +z  =4

3:  -3x+2y-2z=-10

 

First let's try to combine two of the equations.  I'm going to use Eq 1 and Eq 2

1:  x -2y +3z =7
2: 2x +y +z =4

 

Multiply Eq 1 by -2 so that the coefficients of the x's in Eq 1 and Eq 2 are the same magnitude but opposite times.

 

1:  -2x + 4y - 6z = -14

2:   2x +   y + z =    4

 

Add these two equations together, term by term.  We'll call this Eq 4

 

4:         5y - 5z = -10

 

Now let's look at equations 1 and 3.

1:    x -2y +3z =7
3: -3x+2y-2z=-10

 

Multiply Eq 1 by 3 so that the coefficients of the x's in Eq 1 and Eq 3 are the same magnitude but opposite times.

 

1:  3x - 6y + 9z = 21

3: -3x +2y -  2z=-10

 

Add these two equations together, term by term. We'll call this Eq 5

5:        -4y + 7z = 11

 

Now let's look at equations 4 and 5

 

4:  5y  - 5z =-10

5: -4y + 7z = 11

 

In order to eliminate one of the variables, we'll need to adjust both equations.  Multiply Eq 4 by 5 and Eq 5 by 4

4:  20y - 20z = -40

5: -20y + 35z = 55

 

Now add these together term by term

 

15z = 15

z = 1

 

Use this value of z in either Eq 4 or Eq 5 to solve for y

4: 5y - 5z =-10

    5y - 5(1) = -10

5y - 5 = -10

5y = -5

y = -1

 

Now we have values for y and z.  Use these values in either of the original equations to solve for x.

 

1: x -2y +3z =7

x - 2(-1) + 3(1) = 7

x+2+3 = 7

x+5 = 7

x = 2

 

So

x=2

y = -1

z = 1