Download Linear Equations - Math GR. 6-8

Systems of
Linear Equations
Block 44
System of Linear Equations
A system of equations is a set or
collection of equations that you deal
with all together at once.
Linear equations (ones that graph as
straight lines) are simpler than nonlinear equations
The simplest linear system is one
with two equations and two variables.
Graph of a Linear Equation
Graph of y = 3x – 2
x
y
1
1
0
-2
-1
-5
Graph of a Linear Equation
Graph of y = –x – 6
x
y
1
-7
0
-6
-1
-5
System of Linear Equations
Graph of y = 3x – 2 & y = –x – 6
x
y
x
y
1
1
1
-7
0
-2
0
-6
-1
-5
-1
-5
System of Linear Equations
Graph of y = 3x – 2 & y = –x – 6
x
y
x
y
1
1
1
-7
0
-2
0
-6
-1
-5
-1
-5
Solution is (-1, -5)
Practice Solving Systems
of Linear Equations
Solve by Graphing the following systems of linear
equations (see worksheet #1):
#1
Practice Solving Systems
of Linear Equations
Solve by Graphing the following systems of linear
equations (see worksheet #1):
#2
Practice Solving Systems
of Linear Equations
Solve by Graphing the following systems of linear
equations (see worksheet #1):
#3
Practice Solving Systems
of Linear Equations
Solve by Graphing the following systems of linear
equations (see worksheet #1):
#4
Practice Solving Systems
of Linear Equations
Solve by Graphing the following systems of linear
equations (see worksheet #1):
#5
Solving Systems of Linear Equations
Substitution Method:
2x – 3y = –2
4x + y = 24
Choose 2nd equation:
4x + y = 24
Rewrite with single variable: y = 24 – 4x
Substitute into 1st equation: 2x – 3(24 – 4x) = –2
Solving Systems of Linear Equations
Substitution Method:
2x – 3y = –2
4x + y = 24
Simplify:
2x – 72 + 12x = –2
14x – 72 = -2
14x = 70
x=5
Solving Systems of Linear Equations
Substitution Method:
2x – 3y = –2
4x + y = 24
Substitute x = 5 into either equation:
4x + y = 24
4(5) + y = 24
20 + y = 24
y = 24 – 20
y=4
Solving Systems of Linear Equations
Substitution Method:
2x – 3y = –2
4x + y = 24
The solution is the ordered pair (5, 4).
Practice Solving Systems
of Linear Equations
Solve by Substitution the following systems of
linear equations (see worksheet #2):
#1
Practice Solving Systems
of Linear Equations
Solve by Substitution the following systems of
linear equations (see worksheet #2):
#2
Practice Solving Systems
of Linear Equations
Solve by Substitution the following systems of
linear equations (see worksheet #2):
#3
Practice Solving Systems
of Linear Equations
Solve by Substitution the following systems of
linear equations (see worksheet #2):
#4
Practice Solving Systems
of Linear Equations
Solve by Substitution the following systems of
linear equations (see worksheet #2):
#5
Solving an Equation
Addition or Elimination Method:
Example:
x + 6 = 11
-6 -6
x
= 5
Solving Systems of Linear Equations
Addition or Elimination Method-easy:
2x + y = 9
3x – y = 16
Add:
5x = 25
Simplify: x = 5
Substitute: 2(5) + y = 9
10 + y = 9
y = -1
Solving Systems of Linear Equations
Addition or Elimination Method - easy:
2x + y = 9
3x – y = 16
Solution is (5, -1)
Solving Systems of Linear Equations
Addition or Elimination Method – medium:
2x – y = 9
3x + 4y = –14
Multiply 1st by 4: 8x – 4y = 36
8x – 4 y = 36
3x + 4y = –14
Solving Systems of Linear Equations
Addition or Elimination Method – medium:
8x – 4 y = 36
3x + 4y = –14
Multiply 1st by 4: 8x – 4y = 36
Add:
11x = 22
Simplify: x = 2
Substitute: 2(2) – y = 9
4– y= 9
-y = 5 or y = -5
Solving Systems of Linear Equations
Addition or Elimination Method – medium:
2x – y = 9
3x + 4y = –14
Solution is (2, -5)
Solving Systems of Linear Equations
Addition or Elimination Method – hard:
4x – 3y = 25
–3x + 8y = 10
Multiply 1st by 3:
Multiply 2nd by 4:
12x – 9y = 75
-12x + 32y = 40
Solving Systems of Linear Equations
Addition or Elimination Method – hard:
12x – 9y = 75
-12x + 32y = 40
Add:
23y = 115
Simplify: y = 5
Substitute (original equation) : 4x – 3y = 25
4x – 3(5) = 25
4x = 40
Solution is (10, 5)
x = 10
Practice Solving Systems
of Linear Equations
Solve by Addition/Elimination the following
systems of linear equations (see worksheet #3):
#1
Practice Solving Systems
of Linear Equations
Solve by Addition/Elimination the following
systems of linear equations (see worksheet #3):
#2
Practice Solving Systems
of Linear Equations
Solve by Addition/Elimination the following
systems of linear equations (see worksheet #3):
#3
Practice Solving Systems
of Linear Equations
Solve by Addition/Elimination the following
systems of linear equations (see worksheet #3):
#4
Practice Solving Systems
of Linear Equations
Solve by Addition/Elimination the following
systems of linear equations (see worksheet #3):
#5