This document discusses different methods for solving systems of linear equations, including substitution, elimination, and graphing. It provides examples of using substitution and elimination to solve the system 2Y = X + 2, 7X - 12Y = 0. Substitution involves solving one equation for one variable and substituting it into the other, while elimination involves multiplying equations to create common terms that can be eliminated. Graphing involves plotting the lines defined by each equation on a graph and finding their point of intersection.

1. Solving
Systems of
Linear
Equations

2. There are various
methods to solve
systems of linear
equations.
We are going to be
focusing on those
different methods in
this lecture.

3. Let’s first warm-up by doing
some review!
 What is a linear equation?
 What is a system of linear equations?
 What does “solve” mean?
 Which are the dependent and independent variables?

4. What does solving a system
actually do for us?
 Graphically, a solution to a system, gives
us the point at which the two lines intersect
each other.
 Algebraically, a solution to a system means
an x and a y value that satisfy both
equations at the same time

5. The Substitution Method
Steps:
 Solve one of the equations for either x or y
 Substitute the solution from the step above into the second
equation
 Solve this new equation for the remaining variable
 Use the solution from above and solve for the other variable
using one of the original equations (it doesn’t matter which,
you can choose!)

6. Let’s go through an example
together

7. Check for Understanding
 Try:
2Y = X + 2
7X – 12Y = 0
Go through your steps!

8. 2Y = X + 2 (1)
7X – 12Y = 0 (2)
STEP 1: Solve equation (1) for X
X = 2Y – 2 (3)
STEP 2: Substitute equation (3) into equation (2)
7 (2Y - 2) – 12Y = 0 (4)
STEP 3: Solve for Y in equation (4)
Y = 7
STEP 4: Substitute Y = 7 into either equation (1) or (2) to
solve for your value of X
2(7) = X + 2  X = 11
So our solution is (7,12)

9. The Elimination Method
In order to use the elimination method we need to create
variables that have the same coefficient, then we can
eliminate them.
Steps:
 Rearrange your equations (if needed) to line up your
variables in the same order
 Pick a variable (either X or Y) to eliminate
 Multiply or divide by numbers to make sure that both of
those variables have the same coefficient
 Add or subtract them together to eliminate them
 Solve for the remaining variable
 Substitute the solution you just found into either one of the
original equations to find your other variable

11. Check for Understanding
 Try:
2Y = X + 2
7X – 12Y = 0
Go through your steps!
(Yes this is the same example as before! We
are just going to use a different method to get
the same answer)

12. 2Y = X + 2
7X – 12Y = 0
STEP 1: Line up the equations
2Y – X = 2 (1)
-12Y + 7X = 0 (2)
STEP 2: Pick a variable to eliminate (lets pick X)
STEP 3: Multiply equation (1) by 7 so that our X’s have the same
coefficients
14Y – 7X = 14 (3)
-12Y + 7X = 0 (2)
STEP 4: Add the two equations to eliminate the X’s
2Y = 14  Y = 7
STEP 5: Substitute Y = 7 into equation (1)
X= 12
So our solution is (7,11)

13. The Graphing Method
 The graphing method is the most simplistic
method!
Steps:
 Put both your equations into slope-intercept
form
 Graph each line
 Mark where the two lines intersect

14. Watch this video and try this example with our
favorite online-tutorial guy! PatrickJMT

15. Summary:
 Which of these methods were your favorite?
Why?
 If we had three variables and were given
three equations how would you go about
solving that system? Which method do you
think would be easier for a system of three
equations? Four?