3.2 – solving systems of eqs. algebraically
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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. - PowerPoint PPT PresentationTRANSCRIPT
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)