Solving Systems of Equations
by Substitution
(it sounds like really hard math)
Systems of Equations
• A system of equations is a set of two or more
equations that contain two or more variables.
• A solution of a system of equations is a set of
values that are solutions of all the equations.
• If the system has two variables the solution
can be written as an ordered pair. (x,y)
Solving Systems of Equations
•
•
•
•
Isolate like variables in each equation
Set equations equal to each other
Solve for the variable
Substitute the solved variable back into
one of the original equations.
Solving Systems of Equations
Solve each system of equations
Both equations are equal to y so by the transitive
Ex. y = x + 3
property x + 3 is equal to 2x + 5
y = 2x + 5
x + 3 = 2x + 5
-x
-x
3=x+5
-5
-5
-2 = x
y = -2 + 3
y=1
The solution is (-2, 1)
Solve the system of equations
Ex.
3x + 3y = 15
Solve both equations for y
3x – 6y = -12
3x + 3y = 15
3(2) + 3y = 15
3x + 3y = 15
-3x
-3x
3x – 6y = -12
-3x
-3x
3y = 15 – 3x
– 6y = -12 – 3x
/-6
/-6
/3
/3
y=5-x
y = 2 + .5x
6 + 3y = 15
-6
-6
3y = 9
/3
/3
y=3
Both equations are equal to y so by the transitive property 5 – x is equal to 2 + 5x
5 – x = 2 + .5x
+x
+x
5 = 2 + 1.5x
-2
-2
3 = 1.5x
/1.5
/1.5
2=x
The solution is (2,3)
Strange examples
Solve each system of equations
Solve each system of equations
y = 3x + 5
2y = 10x + 20
y = 3x - 10
y = 5x + 10
3x + 5 = 3x - 10
-3x
-3x
5 = -10
No Solution
2y = 10x + 20
/2
/2
y = 5x + 10
y = 5x + 10
So…
y = 5x + 10
y = 5x + 10
5x + 10 = 5x + 10
-5x
-5x
10 = 10
All real numbers are solutions