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 = 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 = 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) = 11

X + 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 solve

Step 3: substitute the solution back into the equation from step 1 and solve

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 = 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 = 4

X = -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