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) + b Equation.

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Presentation transcript:

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) + b Equation #2: y = m(x) + b Step 2: find where the line crosses the y-axis (b) Step 3: determine the slope (m) m = rise / run m = 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 + 5 y = 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 = 6 4x + 2y = 8 (2, 0) 3x + y = 6 – 3x – 3x y = – 3x + 6 4x + 2y = 8 4x + 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 = 32 3x – 6y = 3 (5, 2) 7x + 0 = 35 7x = 35 7x / 7 = 35 / 7 x = 5 4 (5) + 6y = y = y = 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 = – 8 2x + 3y = – 5 (2, – 3) 5x + 6y = – 8 2x + 3y = – 5 – 2 (2x + 3y = – 5) – 4x – 6y = 10 5x + 6y = – 8 – 4x – 6y = 10 x = 2 2x + 3y = – 5 2 (2) + 3y = – y = – 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