Definition
 A system of linear equations, aka linear system,
consists of two or more linear equations with the same
variables.
 x + 2y = 7
 3x – 2y = 5
The solution
 The solution of a system of linear equations is the
ordered pair that satisfies each equation in the system.
 One way to find the solution is by graphing.
 The intersection of the graphs is the solution.
Example
X + 2y = 7
3x – 2y = 5
 Step 1: graph both equations
 Step 2: estimate coordinates of the intersection
 Step 3: check algebraically by subsitution
Types of systems
 Consistent Independent System – has exactly one
solution
*other types to be discussed later
More examples
-5x + y = 0
5x + y = 10
-x + 2y = 3
2x + y = 4
Multi-step problem
 A business rents in line skates ad bicycles. During one
day the businesses has a total of 25 rentals and collects
$450 for the rentals. Find the total number of pairs of
skates rented and the number of bicycles rented.
 Skates - $15 per day
 Bikes - $30 per day
x + y = 25
15x + 30y = 450
 Now find the totals when there were only 20 rentals
and they made $420.
Steps
Step 1: Solve one of the equations
for a variable
Step 2: substitute the expression
in the other equation for the
variable and solve
Step 3: substitute the solution
back into the equation from step
1 and solve
3x – y = -2
X + 2y = 11
3x + 2 = y
X + 2(3x + 2) = 11
X + 6x + 4 = 11
7x = 7
X=1
3(1) + 2 = y
5=y
Solution: (1,5)
More examples
X – 2y = -6
4x + 6y = 4
Y = 2x + 5
3x + y = 10
3x + y = -7
-2x + 4y = 0
Multi-step problem
 A group of friends takes a day-long tubing trip down a
river. The company that offers the tubing trip charges
$15 to rent a tube for a person to use and $7.50 to rent
a tube to carry the food and water in a cooler. The
friends spend $360 to rent a total of 26 tubes. How
many of each type of tube do they rent?
X + y = 26
15x + 7.5y = 360
Elimination Method
2x + 3y = 11
-2x + 5y = 13
Step 1: Add the equations to
eliminate one variable.
(1,3)
Step 2: Solve the resulting
equation for the other
variable.
Step 3: Substitute into
either original equation to
find the value of the other
variable.
8y = 24
8y = 24
Y=3
2x + 3(3) = 11
2x + 9 = 11
2x = 2
X=1
A little twist
Step P: Make Opposite
Step 1: Add
Step 2: Solve
Step 3: Substitute/Solve
4x + 3y = 2
5x +– 3y
-1( -5x
3y =
= -2
2)
-x
=4
X = -4
4(-4) + 3y = 2
-16 + 3y = 2
3y = 18
Y=6
(-4, 6)
Arranging like terms
 If two linear systems are not in the same form you
must rearrange one!
 8x – 4y = -4
 4y = 3x + 14
Examples
4x – 3y = 5
-2x + 3y = -7
-5x – 6y = 8
5x + 2y = 4
3x + 4y = -6
2y = 3x + 6
You try:
7x – 2y = 5
7x – 3y = 4
2x + 5y = 12
5y = 4x + 6