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Honors Algebra 2
Solving Systems of Equations
Name__________________________
Date__________________________
A System of Equations is:
 2 or more equations of lines
 The solution (answer) is the ordered pair (x, y) for when the lines are equal.
Method 1: GRAPHING
 The solution is the point at which the lines __________________________.
EXAMPLE 1:
2x – y = 8
x+y=1

In this example, the system of equations has one solution. We call this system
CONSISTENT and INDEPENDENT.
EXAMPLE 2:
2x + 3y = 6
4x + 6y = 12
EXAMPLE 3:
x – 2y = -6
2x – 4y = 8
Honors Algebra 2
Solving Systems of Equations
Name__________________________
Date__________________________
Examples 2 and 3 represent special cases.
 If the lines are parallel, then there is no solution. We call this INCONSISTENT

If the lines coincide (the lines are equal), then there are infinitely many solutions. This
type of system is called CONSISTENT and DEPENDENT.
PRACTICE: Solve each system by the graphing method.
1) 4x + 4y = 0
x – y = 12
3)
y = 4x – 5
8x – 2y = 20
2) 2x + y = 9
x + 4y = 1
4) 2x – y = 1
3y + 3 = 6x
Honors Algebra 2
Solving Systems of Equations
Method 2: SUBSTITUTION
Name__________________________
Date__________________________
(TRANSFORM ONE EQUATION AND PLUG IT IN)
EXAMPLE 4:
y = 4x
x + 2y = 18
EXAMPLE 5:
-2x + 3y + 22 = 0
x + 4y = 0
Steps:
1) Transform (rewrite) one of the two equations so that it is x = ……. or y = ………
2) Plug in that transformed equation into the untouched equation.
3) Solve for the variable.
4) Use that answer to find the other variable by plugging it in to either of the original equations.
5) Write the answer as an ordered pair (x, y).
6) Check the solution.
7) Classify what type of system it is: Consistent/Independent
(Intersecting Lines)
Consistent/Dependent (Coincide)
Inconsistent (Parallel Lines)
Honors Algebra 2
Solving Systems of Equations
Name__________________________
Date__________________________
PRACTICE: Solve by the substitution method. Classify each system.
5)
x + y = 15
4x + 3y = 38
6) 2x – 3y = 4
x + 4y = -9
METHOD 3: Elimination
Part A: Addition and Subtraction
EXAMPLE 6: Solve the system of equations.
5x – y = 12
3x + y = 4
(If you were to add these 2 equations, which variable would cancel?)
EXAMPLE 7: Solve the system of equations.
6x + 7y = -15
6x – 2y = 12
(If you were to subtract these 2 equations, which variable would cancel?)
STEPS for ADDITION or SUBTRACTION ELIMINATION:
1) Add or subtract the equations in order to cancel one of the variables.
(It all depends on the coefficients and signs of each variable.)
2) Solve the resulting equation for the variable.
3) Use that answer to find the other variable by substituting it back into one of the
original equations.
4) Check your answers.
5) Write the answer as an ordered pair (x, y).
6) Classify the system.
Honors Algebra 2
Solving Systems of Equations
Name__________________________
Date__________________________
Part B: Multiplication with Addition
**Multiplication is used for when the coefficients do not match up for one of the variables.
**Also used for when the coefficients are fractions.
EXAMPLE 8:
4x – 5y = 23
3x + 10y = 31
EXAMPLE 9:
3x + 4y = 2
5x + 9y = 1
EXAMPLE 10:
5
x y 7
3
y 7
x 
4 2
Honors Algebra 2
Solving Systems of Equations
Name__________________________
Date__________________________
Solving Systems of Equations in Three Variables
** Similar to solving a system with 2 variables. There is extra steps because there is an extra
variable. **
** If there are three variables, then there has to be at least 3 equations in order to solve the
system.
x  y  z  0

EXAMPLE 1:
2 x  2 y  3z  5
 x  4 y  3z  11

EXAMPLE 2:
EXAMPLE 3:
2 x  3 y  2 z  14

4 x  2 y  z  15
 x  y  3z  8

 2 x  y  z  11

3 y  z  8
2 z  8
