2. Learning Outcomes
Understand that solutions to a system of two linear equations in
two variables correspond to points of intersection of their
graphs
Solve systems of two linear equations in two variables
algebraically
3. System of Linear
Equations
Simultaneous equations are two or more
algebraic equations that can be solved at the same
time. Linear simultaneous equations with two
unknown variables can be solved by using the
following methods:
Graphical Method
Elimination Method
Substitution Method
4. Graphs of Simultaneous
Equations
1 2 3 4
-4 -3 -2 -1
1
2
3
4
The solutions of simultaneous equations
with two unknown variables correspond
to the point of intersection of their
graphs. -4
-3
-2
-1
Here is a system of two equations:
x + y = 2
y = x + 1
The solution to the simultaneous
equations is x = 0.5, y = 1.5
x + y = 2
y = x + 1
5. Elimination Method
In the elimination method, one variable is eliminated by
adding or subtracting the equations. This leaves one equation
with one variable.
x + 4y = 6
x + 3y = 4
-
y = 2
Subtracting these equations
eliminates the x variable
x + 2y = 6
3x - 2y = 10
+
4x = 16
Adding these equations
eliminates the y variable
6. Exampl
e
1 5x + 2y = 24
3x + 2y = 16
-
2x
x = 4
1
2
Substitute into equation
5(4) + 2y = 24
20 + 2y = 24
2y = 4
y = 2
Solve the equation for y
Look for the same
coefficient. Subtract if
they both have the same
sign
1
x = 4, y = 2
The y variable has been
eliminated, solve for x
= 8
2 2
5x + 2y = 24
7. Exampl
e
2 2x - 2y = - 6
x + 2y = 3
+
3x = - 3
x = - 1
1
2
Substitute into equation
2(-1) - 2y = - 6
-2 - 2y = - 6
- 2y = - 4
y = 2
Solve the equation for y
Look for the same
coefficient. Add if they
have different signs
1
x = -1, y = 2
The y variable has been
eliminated, solve for x
8. More Examples:
3 3x + 2y = 11
3x + y = 7
y = 4
Substitute y = 4
3x + 2y = 11
3x + 2(4) = 11
3x + 8 = 11
3x = 11 – 8
3x = 3
3 3 x = 1
