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 solutionThe 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 systemsTypes of systems  Consistent Independent System – has exactly one solution *other types to be discussed later

More examplesMore examples -5x + y = 0 5x + y = 10 -x + 2y = 3 2x + y = 4

Multi-step problemMulti-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 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) 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 examplesMore examples X – 2y = -6 4x + 6y = 4 Y = 2x + 5 3x + y = 10 3x + y = -7 -2x + 4y = 0

Multi-step problemMulti-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 MethodElimination 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 Y = 3 2x + 3(3) = 11 2x + 9 = 11 2x = 2 X = 1 (1,3)

A little twistA little twist 4x + 3y = 2 5x + 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 y = 2 3y = 18 Y = 6 (-4, 6)

Arranging like termsArranging 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