Systems of Linear Equations!
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Transcript Systems of Linear Equations!
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