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Solving Linear Systems Using Substitution, Elimination, and Graphing to solve a system of equations Table of Contents Review Graphing A Line Define a Linear System Slope Intercept Form Standard Form of a Line Solving a Linear System Parallel Lines Perpendicular Lines Choose between Substitution and Elimination Solving a System using Substitution Solving a System using Elimination Review of Graphing a Line Put equation into slope intercept form Practice – Solve for y: 4x - 3y = -9 Identify the slope and the y-intercept Locate the y-intercept on the y-axis Starting at that point, move up (+) or down(-), then right Show me how Continue Solve for y: 4x - 3y = -9 Step 1: Subtract the X term - 3y = - 4x – 9 Step 2: Divide by the coefficient of y Do NOT divide by y, JUST the coefficient -3y = -4x -9 -3 -3 -3 End result: y = 4x + 3 3 Back Graph a line example ● ● ● Continue Graphing Practice Graph the following equations: 5x – 6y = 8 2x + 4y = -6 y = -3 x + 2 2 Define a Linear System 2 linear equations create a linear system Can be in slope intercept or standard form Y form (Slope Intercept Form) Slope intercept form y=mx+b m is the slope (steepness of line) b is the y-intercept (shows where the line crosses the y axis) Y intercept is also where x = 0 (0,b) Standard Form of a Line Ax + By = C A, B, and C are integers NO FRACTIONS, NO DECIMALS A, B and C can be NEGATIVE Solving a Linear System What does this mean? Locating where the lines cross Where lines share a coordinate (x,y) How many solutions? Parallel lines – no solutions Same equations – infinite solutions Any other lines – one solution Perpendicular lines Parallel Lines Same slope Lines don’t cross – no solution Solve equation – answer is two different numbers EXAMPLE: y=3x+4 y=3x-5 If slope is the same and y-intercept isn’t, the lines are parallel Same Equations Place both into slope intercept form If the equations are identical Lines are located on top of one another End result – ONE line Perpendicular Lines One solution – the lines cross Cross at a 90 degree angle How to tell if two lines are perpendicular? Put both equations in slope intercept form Multiply the slopes If answer = -1, they are perpendicular Choose between Substitution and Linear Combinations (Elimination) Solve with Substitution if: One equation solves for x (i.e. x=) One or both equations are in y form (slope-int) Solve with Linear Combinations (Elimination) if: Both equations are in Standard Form with coefficients other than 0 Steps for Solving a System with Substitution Replace (substitute) one variable, and solve for the other (i.e. replace y and solve for x, or replace x and solve for y) Replace that variable in one equation and solve for the other variable Check your answers in BOTH equations Continue Solving a System with Substitution Part 1 EXAMPLE: 2x-3y=5 y=-2x+1 replace y in first equation with -2x+1 2x - 3 (-2x + 1) = 5 Continue Solving a System with Substitution Part 2 Distribute and combine like terms 2x - 3 (-2x + 1) = 5 2x + 6x - 3 =5 8x-3 =5 Solve for x 8x = 8 x=1 Continue Solving a System with Substitution Part 3 Solve for the second variable Replace x into one equation 2(1) - 3 y = 5 Solve for y 2 - 3y = 5 - 3y = 3 y = -1 The solution is (1, -1) Continue Solving a System with Substitution Part 4 Check your answer in both equations 2x -3y =5 2(1) - 3(-1) = 5 2 + 3 =5 5 =5 y = -2 x +1 -1 = -2 (1) +1 Both answers check -1 = -2 +1 Continue -1 = -1 Solve a System – Elimination Part 1 Be sure both equations are in Standard Form 3x + 5y = 8 -6x – 3y = -2 Solve a System – Elimination Part 2 Multiply one or both equations by a constant END RESULT – coefficients of one variable are + and – of same number 2 (3x + 5y = 8) 6x + 10y = 16 -6x – 3y = -2 -6x – 3y = - 2 Solve an System – Elimination Part 3 Combine the equations 6x + 10y = 16 -6x – 3y = - 2 7y = 14 Divide by coefficient y = 14/7 y=2 Solve an System – Elimination Part 4 Solve for the second variable Replace y into one ORIGINAL equation 3x + 5 (2) = 8 Solve for y 3x + 10 = 8 3x = -2 x = -2/3 The solution is (-2/3, 2) Solve an System – Elimination Part 5 Check your work Replace x and y into both equations 3 x + 5 y =8 -6 x – 3 y = -2 3(-2/3) + 5 (2) = 8 -6(-2/3)-3(2) = -2 -2 + 10 = 8 4 - 6 = -2 8 = 8 -2 = -2 Both solutions check (-2/3, 2)

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posted: | 1/3/2012 |

language: | English |

pages: | 24 |

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