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Study Guide Algebra I Chapter 7 Systems of Equations, pages 374 - 393 As always review your notes and homework. REMEMBER: Solving a system can have 1 of 3 outcomes: 1. A solution (x,y) when the 2 lines intersect 2. No solution when the 2 lines are parallel 3. Infinite solutions when the lines are the same 1. Be able to solve systems of equations by graphing both lines and finding the point of intersection Examples: Solve these equations by graphing y = 2x – 7 & y = -x + 2 y = 2x – 7 m (slope) = y = -x + 2 2 1 −1 m (slope) = 1 € b (y-intercept) = -7 b (y-intercept) = 2 Graph both lines: € 6 4 2 -10 -5 5 10 -2 -4 -6 -8 Where the 2 lines intersect is your solution (3, -1) 2. Be able to solve systems of equations by substitution Example: Solve these 2 equations by substitution 2x – y = 6 x + y = -12 1. Solve 1 of the equations for x or y. In this case it is easier to solve the 2nd equation: x + y = -12 -x -x subtract x from both sides y = -x - 12 2. substitute –x - 12 into the 1st equation for y: 2x –(-x - 12) = 6 2x + x + 12 = 6 3x + 12 = 6 - 12 -12 3x = -6 3 3 x = -2 distribute the – sign combine the x terms solve for x 3. substitute -2 back into the 1st equation to find y y = -x – 12 y = -(-2) – 12 y = 2 – 12 = -10 4. your answer is (-2, -10) ***WRITE YOUR ANSWER AS AN ORDERED PAIR*** 3. Be able to solve linear equations by elimination. Example: solve these equations by elimination: 7x – 8y = 6 4x + y = 9 if you multiply this equation by 8 then the y terms will be opposite and will cancel out 8(4x + y = 9) 32x +8y = 72 multiply ENTIRE equation by 8 Now add the 2 equations together: 32x +8y = 72 1st equation multiplied by 8 + 7x – 8y = 6 copy 2nd equation 39x = 78 ADD 39 39 divide by 39 x=2 solve Once you have a value for x substitute it in either of the ORIGINAL equations and determine the value of y: 4x + y = 9 4(2) + y = 9 substitute 2 for x 8 +y=9 y=1 solve for y (2,1) YOUR ANSWER ***WRITE YOUR ANSWER AS AN ORDERED PAIR*** Example: Solve these equations by elimination: 3x + y = 5 6x + 2y = 10 -2(3x + y = 5) make the y terms opposite by multiplying the 1st equation by -2 -6x - 2y = -10 multiply ENTIRE eq by -2 + 6x + 2y = 10 copy 2nd equation 0 = 0 ADD since 0 = 0 this means the 2 equations are the same line so your answer is SAME LINE or Infinite solutions Example: solve these equations by elimination: 3x - y = -2 -9x + 3y = 3 3(3x - y = -2) make the x terms opposite by multiplying the 1st equation by 3 9x – 3y = -6 + -9x + 3y = 3 0 = -3 multiply ENTIRE eq by 3 copy 2nd equation and ADD since 0 cannot = -3 your answer is NO SOLUTION REMEMBER you can work with EITHER the x or y terms it is your choice. WATCH YOUR NEGATIVE SIGNS !!!!!!!!! REMEMBER you can multiply BOTH equations by numbers to make opposite terms. Example: 6x + 5y = -8 2x - 7y = 32 multiply eq. 1 by 7 multiply eq. 2 by 5 7(6x + 5y = -8) 5(2x - 7y = 32) 42x + 35y = -56 10x – 35y = 160 52x = 104 x=2 6x + 5y = -8 6(2) + 5y = -8 12 + 5y = -8 5y = -20 y = -4 (2, -4) your answer ADD OR 6x + 5y = -8 2x - 7y = 32 -3(2x - 7y = 32) multiply eq 2 by -3 6x + 5y = -8 -6x +21y = -96 26y = -104 y = -4 ADD 2x - 7y = 32 2x -7(-4) = 32 2x + 28 = 32 2x = 4 x=2 (2, -4) your answer 4. Be able to solve word problems using systems of equations To receive full credit you must show all work and have 3 elements in your solution: 1. A let statement 2. 2 equations 3. Your answer must be in a sentence For examples look at notes and homework.