3.1 Graphing Systems of Equations

Algebra IIMrs. Aguirre

Fall 2013

Objective

• You will solve a system of equations by graphing.

Application

• Reliable Rentables rents moving trucks for $40 per day plus 35¢ per mile driven. The Mover’s Helper rents trucks for $36 a day plus 45¢ per mile driven. When is the total cost of a day’s rental the same for both companies? When is it better to rent from Reliable Rentals?

Now what?

• Let d = the total cost of a day’s rental.

• Let m = the miles driven.

Write the following equations:

d = 40 + 0.35m (Total cost of renting from RR)

d = 36 + 0.45m (Total cost of renting from MH)

Now graph the two equations• By graphing these two

equations, we can see how the rental rates compare. Each point on a line has coordinates that satisfy the equation of the line. Since (40, 54) is on both lines, it satisfies both equations. So, if you rent a truck from either company and drive 40 miles, the price will be $54. It is better to rent from Reliable Rentables when you drive more than 50 miles.

• Together the equations d = 40 + 0.35m and d = 36 + 0.45m

are called a system of equations. The solution of this system is (40, 54) where they meet.

Using TI Calculator• Make sure they are in slope

intercept form (y=).

• Fix window. (Zoom 3 until you can see both lines and their intersection!)

• 2nd Trace (Calc) 5 enter

Using TI Calculator

• Enter 3 times.

• Intersection point on bottom of screen.

Ex. 1: Solve this system of equations by graphing:

The slope-intercept form of

x + y = 4 is y = -x +4

932

4

yx

yx

The slope-intercept form of

2x + 3y = 9 is 33

2 xy

Since the two lines have different slopes, the graphs of the equations are intersecting lines.

They intersect at (3, 1).

The solution of the system is (3, 1).

Ex. 2: Solve this system of equations by graphing:

The slope-intercept form of

2y + 3x = 6 is

1264{

632{

xy

xy

The slope-intercept form of

4y + 6x = 12 is 32

3 xy

Since the two lines have the same slopes and y-intercept, the graphs are the same line.

There are infinitely many solutions to this system.

32

3 xy

Solution set is {(x, y) | 2y + 3x = 6}

Ex. 3: The plumbing problem • Ex. 3: Perry’s Plumbing charges $35 for

any service call plus an additional $40 an hour for labor. A service call from Rapid Repair Plumbing costs $45 plus an additional $40 an hour for labor. When is the total price for a service call the same for both companies? When is it better to use Perry’s Plumbing?

• Let h represent the hours of labor and p represent the total price of the repair. Write and graph a system of equations.

p = 40h + 35 (Total price from Perry’s)

p = 40h + 45 (Total price from RR)

No points of intersection! No solution! Since they never intersect, the total price from the two companies is never the same. It is always less expensive to use Perry’s

Food for thought . . .

• A system with no solutions like Example 3 is called an inconsistent system. A system that is consistent has lines that are not parallel.

Summary Chart of possibilities for graphs of two linear equations in two variables

Graphs of Equations

Slopes of Lines Name of Systems of Equations

Number of Solutions

Lines intersect Different slopes Consistent & independent

one

Lines coincide Same slope,

Same intercepts

Consistent and dependent

Infinite

Lines parallel Same slope, different

intercepts

Inconsistent none

Chalkboard Example: Solve this system of equations by graphing:

The slope-intercept form of

x + y = 5 is

2023{

5{

yx

yx

The slope-intercept form of

3x - 2y = 20 is 102

3 xy

Since the two lines do not have the same slope, they intersect.

5 xy

They intersect at (6, -1).

The solution of the system is (6, -1).

Chalkboard Example: Solve this system of equations by graphing:

The slope-intercept form of

Y = -3x +5 is

1539{

53{

yx

xy

The slope-intercept form of

9x + 3y = 15 is 53 xySince the two lines have the same slope and y-intercept, their graphs are the same line. Any ordered pair on the line will satisfy both equations.

53 xy

There are infinitely many solutions to this system.

Solution set is {(x, y) | y = -3x + 5}