Transcript Document

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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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 Method
Ex. 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)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
x = -2(-7) + 8
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
x = -2(-7) + 8
x = 14 + 8
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
x = -2(-7) + 8
x = 14 + 8
x = 22
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
x = -2(-7) + 8
x = 14 + 8
x = 22
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
x = -2(-7) + 8
x = 14 + 8
x = 22
(22
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
Substitute into equation from 1) and solve for x.
x = -2y + 8
x = -2(-7) + 8
x = 14 + 8
x = 22
(22
1)
2)
3)
Solve 1st eq. for variable (whichever is easiest)
x + 2y = 8
- 2y - 2y
x = -2y + 8
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
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 Method
Ex. 2 Use the elimination method to solve
the system of equations.
Elimination Method
Ex. 2 Use the elimination method to solve
the system of equations.
a.
4a + 2b = 15
2a + 2b = 7
Elimination Method
Ex. 2 Use the elimination method to solve
the system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) Make numbers of 1 of the variables the
same number with opposite signs, then add
the equations together
Elimination Method
Ex. 2 Use the elimination method to solve
the system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) Make numbers of 1 of the variables the
same number with opposite signs, then add
the equations together
4a + 2b = 15
Elimination Method
Ex. 2 Use the elimination method to solve
the system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 Method
Ex. 2 Use the elimination method to solve
the system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 Method
Ex. 2 Use the elimination method to solve
the system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 Method
Ex. 2 Use the elimination method to solve the
system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 + 0 = 8
Elimination Method
Ex. 2 Use the elimination method to solve the
system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 Method
Ex. 2 Use the elimination method to solve the
system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 Method
Ex. 2 Use the elimination method to solve the
system of equations.
a.
4a + 2b = 15
2a + 2b = 7
1) 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 Method
Ex. 2 Use the elimination method to solve the system of
equations.
a.
4a + 2b = 15
2a + 2b = 7
1)
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 Method
Ex. 2 Use the elimination method to solve the system of
equations.
a.
4a + 2b = 15
2a + 2b = 7
1)
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 Method
Ex. 2 Use the elimination method to solve the system of
equations.
a.
4a + 2b = 15
2a + 2b = 7
1)
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 Method
Ex. 2 Use the elimination method to solve the system of
equations.
a.
4a + 2b = 15
2a + 2b = 7
1)
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 Method
Ex. 2 Use the elimination method to solve the system of
equations.
a.
4a + 2b = 15
2a + 2b = 7
1)
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 = 45
1) Make numbers of 1 of the variables the
same number with opposite signs, then add
the equations together
b.
3x – 7y = -14
5x + 2y = 45
1) 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 = 45
1) 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 = 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
b.
3x – 7y = -14
5x + 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
b.
3x – 7y = -14
5x + 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
b.
3x – 7y = -14
5x + 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 = -14
5x + 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 = -14
5x + 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)