Systems of Linear Equations!
By graphing
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 = 73x – 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 = 05x + y = 10
-x + 2y = 32x + 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 = 2515x + 30y = 450
Now find the totals when there were only 20 rentals and they made $420.
Solve by Substitution
Steps
Step 1: Solve one of the equations for a variable
3x – y = -2X + 2y = 11
3x + 2 = y
X + 2(3x + 2) = 11X + 6x + 4 = 117x = 7X = 1
3(1) + 2 = y5 = y
Solution: (1,5)
Step 2: substitute the expression in the other equation for the variable and solveStep 3: substitute the solution back into the equation from step 1 and solve
More examples
X – 2y = -64x + 6y = 4
Y = 2x + 53x + 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 = 2615x + 7.5y = 360
Elimination 7.3
Elimination Method
Step 1: Add the equations to eliminate one variable.
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.
2x + 3y = 11-2x + 5y = 13
8y = 24
8y = 24 Y = 3
2x + 3(3) = 112x + 9 = 112x = 2X = 1
(1,3)
A little twist
4x + 3y = 25x + 3y = -2-1( )4x + 3y = 2-5x – 3y = 2
-x = 4X = -4
4(-4) + 3y = 2
Step P: Make Opposite
Step 1: Add
Step 2: Solve
Step 3: Substitute/Solve
-16 + 3y = 23y = 18Y = 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 = 85x + 2y = 4
3x + 4y = -62y = 3x + 6
You try: 7x – 2y = 57x – 3y = 4
2x + 5y = 125y = 4x + 6