This document discusses three methods for solving systems of equations: graphing, substitution, and elimination. The graphing method involves finding the intersection of two graphs. Substitution involves rewriting one variable in terms of the other. Elimination involves canceling out a variable by making coefficients of that variable equal between equations. While substitution and elimination work best when equations are already in a certain form, any method can be used to solve systems as needed by manipulating the equations.

1. Systems of Equations

2. Graphing Method
● Why Graphing?
- The fundamental idea: finding the intersection between two graphs
- Graphing helps a lot when dealing with inequalities involving systems of
equations.

3. Graphing Method (cont.)
● Terminologies
- Slope: y = mx+b, also known as rise over run
- X-intercept: the point where y = 0
- Y-intercept: the point where x = 0

4. Graphing Method (cont.)
Checking Your Understanding
You are asked to graph a systems of equations
y = 2x-2 and y = 3x+6.
Based on our lesson so far, what are some of the
things you have to know in order to graph the system?

5. Graphing Method (Cont.)

6. Substitution Method
● What is Substitution?
- A method where you rewrite a variable in terms of the other
● When Do I Use Substitution?
- IDEALLY when one of the functions is given in terms of the other.
- Substitution can be used for any given functions regardless
● Can you Elaborate a Little More?
Ex. Find the solution for 3x-7y=-14 and x=2y-3.
- Notice how the bottom function can directly substitute the x for the top?

7. Substitution Method (cont.)
● Terminologies
- Substitution: Replacing one of the variables in terms of the other
- Think of it this way: It is similar to substituting a sports athlete during a
game!

8. Substitution Method (cont.)
Checking Your Understanding
Which pair of functions is more appropriate for substitution
method? Why?
A) -5x+7y-10 = 0 and 7x+9y = 60
B) y = 11x-5 and x+y = 2
Critical Question: What are some of the reasons why some functions are
more difficult to solve by using substitution method?

9. Substitution Method (cont.)

10. Elimination Method
● What is Elimination?
- A method where you cancel out one of the variables by making one of the coefficients’ absolute values
equal to each other’s
● When Do I Use Elimination?
- IDEALLY when the coefficients of the corresponding variables of both functions are equal and are
given in the form of Ax+By=C
- Just like substitution, elimination can be used for any give functions regardless
● Can You Elaborate A Little More?
Ex. Find the solution for the system 3x-4y = -5 and -10x+4y = 12.
- Notice how both functions have the same coefficients for y, which is 4, and are written in the form of
Ax+By=C.

11. Elimination Method (cont.)
● Terminologies
- Elimination: Canceling out one of the
variables that have the same absolute value
for the coefficients either by addition or
subtraction
- Coefficient: The number in front of the
variable (ie. Coefficient of 5x is 5)

12. Elimination Method (cont.)
Checking Your Understanding
Which pair of functions is the most suitable for elimination method? Why?
A) x-2y = -3 and 7y = 5x+8
B) y = 9x-1 and x = 27y+16
C) 12x+20y = 50 and 15x-20y = 30
Critical Question: What are some of the reasons why some functions are more
difficult to solve by elimination method?

13. Elimination Method (cont.)

14. Your Decision
● Notice the usage of the word IDEALLY
- Very few ideal scenarios for substitution and elimination methods
- Manipulating functions is required most of the time before you can apply any of
the methods
● Any of the three methods can be used to solve systems of equations
with varying effectiveness depending on the functions
- All up to you on how to approach them!