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I. SOLVING BY GRAPHING

Solution(s) Examples of Graphs Classifications

*one solution

has exactly one intersection

Consistent & independent

*infinitely many solutions *Lines have the same slope (m) and y-intercept (b)

over lapping lines (coincide) Consistent & dependent

*No solution *Lines have the same slope (m) only

No intersection Inconsistent

EXAMPLES: SOLVE BY GRAPHING

1) y = -2x 2) y = 3x + 2 3) x + y = 6

y = x + 3 y = 3x – 3 -2x – 2y = -12

USING YOUR CALCULATOR TO GRAPH:

Hotel A charges $70 per night and a one-time $5 facility fee. Hotel B charges $65 per night and a $20 one-time facility fee. After how many nights will the cost of the two hotels be the same?

If you are only staying 2 nights, which would be a better deal?

If you are staying 5 nights, which would be a better deal?

Write equations for both hotels and graph to find the answer.

II. SUBSTITUTION METHOD

Step 1) Solve one equation for one of its variables.

Step 2) Substitute the expression from Step 1 into the other equation and solve for the other variable.

Step 3) Substitute the value from Step 2 into the equation from Step 1 and solve.

EXAMPLES TO FOLLOW…..

EXAMPLE 1) Solve the system by substitution: 6x + 3y = 12 3x + y = 5

EXAMPLE 2: Solve the system by substitution: 2x + y = 4 3x – 5y = 6

EXAMPLE 3: Solve the system by substitution: 3x + 6y = 3 x – 2y = 5

III. ELIMINATION METHOD

Step 1) Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables.

Step 2) Add the equations together and solve for the remaining variable.

Step 3) Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

EXAMPLES TO FOLLOW….

EXAMPLE 1) Solve the system by elimination: 7x + 2y = -5 3x – 4y = -7

EXAMPLE 2) Solve by elimination method: 2x – 3y = 3 4x – 5y = 9

EXAMPLE 3) Solve by elimination method: 3x – y = 5 6x – 2y = 10

EXAMPLE 4) Solve by elimination method: x – 2y = 5 4x – 8y = -3

WORD PROBLEM: Your school sells short sleeve t-shirts that cost the school $5 each and are sold for $8 each. Long sleeve t-shirts cost the school $7 each and are sold for $13 each. The school spends $2450 on t-shirts and sells all of them for $4325. How many of each type of t-shirt are sold?

Define variables:

Write equations:

Solve: