3.2 – Solving Systems of Eqs. Algebraically

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution Method

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

½x – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

½x – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

½x – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8 ½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8 ½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8

½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

½x – y = 18

½(-2y + 8) – y = 18

-y + 4 – y = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8 ½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8 ½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14

3.2 – Solving Systems of Eqs. Algebraically• Recall that when solving graphically, solution is point of

intersection.

Substitution MethodEx. 1 Use substitution to solve the system of equations.

x + 2y = 8 ½x – y = 18

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x.

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

½x – y = 18

½(-2y + 8) – y = 18

-y + 4 – y = 18

-2y + 4 = 18

-2y = 14

y = -7

3) Substitute into equation from 1) and solve for x.

x = -2y + 8

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8

1) Solve 1st eq. for variable (whichever is easiest)

x + 2y = 8

- 2y - 2y

x = -2y + 8

2) Substitute in and solve for other variable!

½x – y = 18

½(-2y + 8) – y = 18

-y + 4 – y = 18

-2y + 4 = 18

-2y = 14

y = -7

3) Substitute into equation from 1) and solve for x.

x = -2y + 8

x = -2(-7) + 8

x = 14 + 8

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22 (22

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22 (22

1) Solve 1st eq. for variable (whichever is easiest) x + 2y = 8

- 2y - 2y x = -2y + 8

2) Substitute in and solve for other variable! ½x – y = 18

½(-2y + 8) – y = 18-y + 4 – y = 18-2y + 4 = 18 -2y = 14 y = -7

3) Substitute into equation from 1) and solve for x. x = -2y + 8 x = -2(-7) + 8 x = 14 + 8 x = 22 (22,-7)

Elimination Method

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.

a. 4a + 2b = 15

2a + 2b = 7

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.

a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.

a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

4a + 2b = 15

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.

a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

4a + 2b = 15

(-1)[2a + 2b = 7]

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

Elimination MethodEx. 2 Use the elimination method to solve

the system of equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

Elimination MethodEx. 2 Use the elimination method to solve the

system of equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same

number with opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -72a + 0 = 8

Elimination MethodEx. 2 Use the elimination method to solve the

system of equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same

number with opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8

Elimination MethodEx. 2 Use the elimination method to solve the

system of equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same

number with opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

Elimination MethodEx. 2 Use the elimination method to solve the

system of equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same

number with opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

2) Plug 4 into first eq. and solve for b.

Elimination MethodEx. 2 Use the elimination method to solve the system of

equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same number with

opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

2) Plug 4 into first eq. and solve for b. 4(4) + 2b = 15

Elimination MethodEx. 2 Use the elimination method to solve the system of

equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same number with

opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

2) Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15

Elimination MethodEx. 2 Use the elimination method to solve the system of

equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same number with

opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

2) Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15 2b = -1

Elimination MethodEx. 2 Use the elimination method to solve the system of

equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same number with

opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

2) Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15 2b = -1

b = -½

Elimination MethodEx. 2 Use the elimination method to solve the system of

equations.a. 4a + 2b = 15

2a + 2b = 71) Make numbers of 1 of the variables the same number with

opposite signs, then add the equations together

4a + 2b = 15-2a - 2b = -7

2a = 8 a = 4

2) Plug 4 into first eq. and solve for b. 4(4) + 2b = 15 16 + 2b = 15 2b = -1

b = -½, So the lines intersect at (4, -½)

b. 3x – 7y = -14

5x + 2y = 45

b. 3x – 7y = -14

5x + 2y = 451) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

b. 3x – 7y = -14

5x + 2y = 451) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

3x – 7y = -14

5x + 2y = 45

b. 3x – 7y = -14

5x + 2y = 451) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

(2)[3x – 7y = -14]

(7)[5x + 2y = 45]

b. 3x – 7y = -14

5x + 2y = 451) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

6x – 14y = -28

35x + 14y = 315

b. 3x – 7y = -14

5x + 2y = 451) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

6x – 14y = -28

35x + 14y = 315

41x = 287

b. 3x – 7y = -14

5x + 2y = 451) Make numbers of 1 of the variables the

same number with opposite signs, then add the equations together

6x – 14y = -28

35x + 14y = 315

41x = 287

x = 7

b. 3x – 7y = -145x + 2y = 45

1) Make numbers of 1 of the variables the same number with opposite signs, then add the equations together

6x – 14y = -28 35x + 14y = 315

41x = 287 x = 7

2) Plug 7 into first eq. and solve for y.

b. 3x – 7y = -145x + 2y = 45

1) Make numbers of 1 of the variables the same number with opposite signs, then add the equations together

6x – 14y = -28 35x + 14y = 315

41x = 287 x = 7

2) Plug 7 into first eq. and solve for y.*Should get y = 5

b. 3x – 7y = -145x + 2y = 45

1) Make numbers of 1 of the variables the same number with opposite signs, then add the equations together

6x – 14y = -28 35x + 14y = 315

41x = 287 x = 7

2) Plug 7 into first eq. and solve for y.*Should get y = 5, so (7,5)