1. Substitution Method & Elimination Method
Systems of Equations
Substitution Method
& Elimination Method
copyright © 2011 by Lynda Aguirre

2. Systems of Equations 2 Equations in 2 variables
Substitution Method
Systems of Equations 2 Equations in 2 variables
copyright © 2011 by Lynda Aguirre

3. Substitution Method
This method takes one equation and substitutes it into the other one.
Why are we doing this? To get an equation with only one variable (unknown value) in it.
Step 1: Solve either equation for x or y (this will be the “original” equation)
Sometimes an equation already has an equation that is solved for x or y.
Call this equation the “original”
Step 2: Replace x or y in the “other” equation with the value from the “original” equation.
Now we have an equation with only one variable in it.
Step 3: Solve for the remaining variable (in this case: x)
Add Like Terms and Isolate x.
This gives us one variable (x=4), now we need to find the other one (y).
copyright © 2011 by Lynda Aguirre

4. Substitution Method Part II: Find the other variable
In steps 1-3, we plugged the 2nd equation into the 1st and found x=4.
Step 4: Plug the value from step 3 (x=4) into the “original” equation
Step 5: Solve for the remaining variable (y).
This gives us both values which we list as a coordinate
Solution: (4, 1)
copyright © 2011 by Lynda Aguirre

5. Substitution Method
Step 1: Solve either equation for x or y (your choice)
1st equation is the “original” equation
My choice: Solve the 1st equation for y:
2nd equation (the “other” equation)
Step 2: Replace the variable in the “other” equation with the value from the “original” equation
1st equation:
The variable
The value of y
2nd equation:
Step 3: Solve for the remaining variable (in this case, solve for x)
Add Like Terms and Isolate x.
This gives us one value (4, ___) Now we need to find the “y”
copyright © 2011 by Lynda Aguirre

6. Substitution Method Part II: Find the other variable
Step 4: Plug the value from step 3, (x=4), into the “original” equation
The original equation
Step 5: Solve for the remaining variable (y).
This gives us both values which we list as a coordinate
Solution: (4, 2)
copyright © 2011 by Lynda Aguirre

7. Substitution Method Solution: (6, 0) Try this one on your own STEPS
Try this one on your own
Solution: (6, 0)
STEPS
1) Solve one equation for x or y, label it “original” 2) Plug “original” into the “other” equation 3) Solve for 1st variable 4) Plug 1st variable into the “original” equation
5) Solve for 2nd variable
6) Write the solution (x, y) Note: if the problem has letters other than x and y in it,
put them in alphabetical order
copyright © 2011 by Lynda Aguirre

8. Substitution Method: steps
1) Solve one equation for x or y, label it “original” 2) Plug “original” into the “other” equation 3) Solve for 1st variable 4) Plug 1st variable into the “original” equation
5) Solve for 2nd variable
6) Write the solution (x, y) Note: if the problem has letters other than x and y in it,
put them in alphabetical order
Things to note: --In step 1, if you choose to solve for a variable with a coefficient, you will create fractions.
--You must substitute into one equation in step 2 and then the other one in step 4
--You can check your answers by plugging the numbers (x,y) into BOTH equations
--Sometimes step 1 is not necessary if one of the equations is already solved for x or y
copyright © 2011 by Lynda Aguirre

9. copyright © 2011 by Lynda Aguirre
copyright © 2011 by Lynda Aguirre

10. Dependent and Inconsistent Systems of Equations
All the examples up to this point were systems of equations that (if graphed) cross at a single point.
Lines that cross at a point (x, y) are “Consistent Systems”.
But it is possible for two lines to be parallel (i.e. they never cross)
A system of parallel Lines is called an “Inconsistent System”
OR Two lines could represent the same line graphed twice (i.e. one on top of the other, so they intersect at every point)
The same line graphed twice is called a
“Dependent System”
copyright © 2011 by Lynda Aguirre

11. Types of Systems and Solutions
Type of System
Solution
Graph
Consistent
(x,y)
Two lines that cross
Inconsistent
No solution
Parallel lines
Dependent
An Infinite Number of Solutions
Same line twice (looks like one line)
copyright © 2011 by Lynda Aguirre

12. copyright © 2011 by Lynda Aguirre
Solution: no solution TOS: Inconsistent
Solution: no solution TOS: Inconsistent
Solution: An infinite number of solutions TOS: Dependent
Solution: An infinite number of solutions TOS: Dependent
copyright © 2011 by Lynda Aguirre

13. Systems of Equations 2 Equations in 2 variables
Elimination Method
Systems of Equations 2 Equations in 2 variables
copyright © 2011 by Lynda Aguirre

14. Elimination (or Addition) Method
This method takes one equation and adds it to the other one.
Why are we doing this? To get an equation with only one variable (unknown value) in it.
Sometimes one or both equations are already in the correct format
Step 2: Draw a line underneath and add the like terms (straight down). One should cancel out.
Now we have an equation with only one variable in it.
Solution:
(4 , 2)
Step 3: Solve for the remaining variable (in this case: x)
Step 4: Substitute this value into “either” of the original equations
copyright © 2011 by Lynda Aguirre

15. Elimination (or Addition) Method
Step 1: Put both equations into General Form
Sometimes one or both equations are already in the correct format
Step 1a: If necessary, multiply one equation (or both) by a number and/or a negative sign so x’s or y’s will cancel (i.e. equal zero)when added
My choice: Make the x’s cancel
Now the x’s have the same number and different signs
Step 2: Draw a line underneath and add the like terms (straight down). One should cancel out.
Now we have an equation with only one variable in it.
Step 3: Solve for the remaining variable (in this case: y)
Solution:
( -4, 7)
Step 4: Substitute this value into “either” of the original equations
copyright © 2011 by Lynda Aguirre

16. Elimination (or Addition) Method
Try this one on your own
Solution: (-4, 7)
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre

17. Elimination (or Addition) Method
Try this one on your own
Solution: (-4, 1)
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre

18. Elimination (or Addition) Method
Try this one on your own
Solution: (-1, -6)
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre

19. Elimination (or Addition) Method
Try this one on your own
Solution: (-8, -2)
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre

20. Elimination Method: steps
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down)
Step 3: Solve for the remaining variable Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
Things to note: --You can check your answers by plugging the numbers (x,y) into BOTH equations
--Sometimes step 1 is not necessary if the equations are already in General Form --If there are fractions in either equation, multiply by the LCD to get rid of them --If there are decimals in either equation, multiply by a power of 10 (10, 100, 1000,…)
copyright © 2011 by Lynda Aguirre

21. copyright © 2011 by Lynda Aguirre
copyright © 2011 by Lynda Aguirre

22. copyright © 2011 by Lynda Aguirre
Using the Elimination Method, name the Solution and the Type of System (TOS)
Solution: no solution TOS: Inconsistent
Solution: no solution TOS: Inconsistent
Solution: An infinite number of solutions TOS: Dependent
Solution: An infinite number of solutions TOS: Dependent
copyright © 2011 by Lynda Aguirre