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6-1 System of Equations ( Graphing ):

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

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6-1 System of Equations ( Graphing ):

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  1. 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

  2. 6-1 Graphing Possible Solutions: Only One Infinite No Solution

  3. Slope-Intercept Form y = m(x) + b m = rise / run m = y-axis / x-axis y = -3x + 5 y = x - 3 Graphing ( x , y ) (2 , -1) + Y - X + X - Y

  4. Slope-Intercept Form y = m(x) + b m = rise / run m = y-axis / x-axis Graphing + Y - X + X - Y

  5. 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.

  6. 3x + y = 64x + 2y = 8 (2, 0) 3x + y = 6 – 3x – 3xy = – 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

  7. POSSIBLE SOLUTIONS Only One (x, y) = crossed lines No Solution (answers don’t equal) = parallel lines Infinite Solutions (answers are equal) = stacked lines

  8. 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 6-3 Elimination (Addition & Subtraction)

  9. 4x + 6y = 323x – 6y = 3 Same Signs - SUBTRACT Opposite Signs + ADD (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 Add / Subtract

  10. POSSIBLE SOLUTIONS Only One (x, y) = crossed lines No Solution (answers don’t equal) = parallel lines Infinite Solutions (answers are equal) = stacked lines

  11. 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). 6-4 Elimination (Multiplication)

  12. 5x + 6y = – 82x + 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= – 5 4 + 3y = – 5 – 4 – 4 3y = – 9 3y / 3 = – 9 / 3 y = – 3 Multiplication

  13. POSSIBLE SOLUTIONS Only One (x, y) = crossed lines No Solution (answers don’t equal) = parallel lines Infinite Solutions (answers are equal) = stacked lines

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