6-1 System of Equations (Graphing):

Step 1: both equations MUST be in slope intercept form before you can graph the lines

Equation #1: y = m(x) + bEquation #2: y = m(x) + b

Step 2: find where the line crosses the y-axis (b)

Step 3: determine the slope (m)m = rise / runm = y-axis / x-axis

Step 4: graph each equation

6-1 Graphing Possible Solutions: Only One Infinite No Solution

Graphing

+ Y

- Y

- X + X

Slope-Intercept Form

y = m(x) + b

m = rise / run

m = y-axis / x-axis

y = -3x + 5y = x - 3

( x , y )(2 , -1)

+ Y

- Y

- X + X

Slope-Intercept Form

y = m(x) + b

m = rise / run

m = y-axis / x-axis

Graphing

6-2 Solving Systems (Substitution)

Step 1: Solve an equation to one variable.

Step 2: Use the common variable and substitute the expression into the other equation.

Step 3: Solve for the only variable left in the equation to find its value.

Step 4: Plug the new value back into one of the original equations to find the other value.

3x + y = 64x + 2y = 8 (2, 0)

3x + y = 6 – 3x – 3xy = – 3x + 6

4x + 2y = 84x + 2(– 3x + 6) = 8 4x – 6x + 12 = 8

– 12 – 12 – 2x = – 4

– 2x / – 2 = – 4 / – 2 x = 2

3x + y = 6 3(2) + y = 6 – 6 – 6 y = 0

Substitution

POSSIBLE SOLUTIONS

1) Only One (x, y) = crossed lines

2) No Solution (answers don’t equal) = parallel lines

3) Infinite Solutions (answers are equal) = stacked lines

6-3 Elimination (Addition & Subtraction)

• Step 1: Line up the equations so the matching terms are in line.

• Step 2: Decide whether to add or subtract the equations to get rid of one variable, then solve.

• Step 3: Substitute the solved variable back into one of the original equations, then write the ordered pair (x, y).

Same Signs - SUBTRACT

Opposite Signs + ADD

4x + 6y = 323x – 6y = 3 (5, 2)

7x + 0 = 35 7x = 35

7x / 7 = 35 / 7 x = 5

4 (5) + 6y = 32 20 + 6y = 32 - 20 - 20 6y = 12

6y / 6 = 12 / 6 y = 2

Same Signs - SUBTRACT

Opposite Signs + ADD

Add / Subtract

POSSIBLE SOLUTIONS

1) Only One (x, y) = crossed lines

2) No Solution (answers don’t equal) = parallel lines

3) Infinite Solutions (answers are equal) = stacked lines

6-4 Elimination (Multiplication)

• Step 1: Line up the equations so the matching terms are in line.

• Step 1.5 (new): Multiply at least one equation to get two equations containing opposite terms (example + 6y and – 6y).

• Step 2: Decide whether to add or subtract the equations to get rid of one variable, then solve.

• Step 3: Substitute the solved variable back into one of the original equations, then write the ordered pair (x, y).

5x + 6y = – 82x + 3y = – 5 (2, – 3)

5x + 6y = – 82x + 3y = – 5

– 2 (2x + 3y = – 5)– 4x – 6y = 10

5x + 6y = – 8– 4x – 6y = 10

x = 2

2x + 3y = – 5 2 (2) + 3y = – 5 4 + 3y = – 5 – 4 – 4 3y = – 9

3y / 3 = – 9 / 3 y = – 3

Multiplication

POSSIBLE SOLUTIONS

1) Only One (x, y) = crossed lines

2) No Solution (answers don’t equal) = parallel lines

3) Infinite Solutions (answers are equal) = stacked lines