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BY GRAPHING Y = 2X + 1 Y = -X + 4 (1,3) IS THE SOLUTION Graphing is not the only way to solve a system of equations. It is not really the best way because it has to be graphed perfectly and some answers are not integers. SOOOO We need to learn another way!!!! Example #1: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Simplify and solve the equation. 7x = -21 x = -3 y = 4x 3x + y = -21 Step 3: Substitute what you solved into Step 1 y = 4x y = 4(-3) y = -12 Solution to the system is (-3, -12). y = 4x 3x + y = -21 Step 4: Check the solution in both equations. Solution to the system is (-3,-12). y = 4x -12 = 4(-3) -12 = -12 3x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21 -21= -21 Example #2: Solve using “Substitution” x + y = 10 5x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2 x + y = 10 5x – y = 2 Simplify and solve the equation. 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x=2 x + y = 10 5x – y = 2 Step 3: Substitute back what you solved into step 1 y = -x + 10 y = -(2) + 10 y=8 Solution to the system is (2,8). Like variables must be lined under each other. We need to eliminate (get rid of) a variable. The x’s will be the easiest. So, we will add the two equations. Solve: by ELIMINATION x + y = 12 -x + 3y = -8 Divide by 4 4y = 4 y=1 THEN---- Substitute your answer into either original equation and solve for the second variable. X +Y = 12 x + 1 = 12 -1 -1 x = 11 (11,1) Answer Now check our answers in both equations------ X + Y =12 11 + 1 = 12 12 = 12 -x + 3y = -8 -11 + 3(1) = -8 -11 + 3 = -8 -8 = -8 Like variables must be lined under each other. We need to eliminate (get rid of) a variable. The y’s be will the easiest.So, we will add the two equations. Solve: by ELIMINATION 5x - 4y = -21 -2x + 4y = 18 Divide by 3 3x = -3 x = -1 THEN---- Substitute your answer into either original equation and solve for the second variable. 5X - 4Y = -21 5(-1) – 4y = -21 -5 – 4y = -21 5 5 -4y = -16 y=4 (-1, 4) Now check our answers in both equations-----Answer 5x - 4y = -21 5(-1) – 4(4) = -21 -5 - 16 = -21 -21 = -21 -2x + 4y = 18 -2(-1) + 4(4) = 18 2 + 16 = 18 Like variables must be lined under each other. We need to eliminate (get rid of) a variable. The y’s will be the easiest. So, we will add the two equations. Solve: by ELIMINATION 2x + 7y = 31 5x - 7y = - 45 Divide by 7 7x = -14 x = -2 THEN---- Substitute your answer into either original equation and solve for the second variable. 2X + 7Y = 31 2(-2) + 7y = 31 -4 + 7y = 31 4 4 7y = 35 y=5 (-2, 5) Now check our answers in both equations-----Answer 2x + 7y = 31 2(-2) + 7(5) = 31 -4 + 35 = 31 31 = 31 5x – 7y = - 45 5(-2) - 7(5) = - 45 -10 - 35 = - 45 - 45 =- 45 Like variables must be lined under each other. We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If one of the x’s was negative, it would be eliminated when we add. So we will multiply one equation by a – 1. Solve: by ELIMINATION x + y = 30 x + 7y = 6 X + Y = 30 X + Y = 30 ( X + 7Y = 6 ) -1 -X – 7Y = - 6 -6Y = 24 Now add the two equations and solve. -6 -6 Y=-4 THEN---- Substitute your answer into either original equation and solve for the second variable. X + Y = 30 X + - 4 = 30 4 4 X = 34 (34, - 4) Now check our answers in both equations-----Answer x + y = 30 34 + - 4 = 30 30 = 30 x + 7y = 6 34 + 7(- 4) = 6 34 - 28 = 6 6=6 Like variables must be lined under each other. We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2. Solve: by ELIMINATION x+ y=4 2x + 3y = 9 ( X + Y = 4 ) -2 2X + 3Y = 9 -2X - 2 Y = - 8 2X + 3Y = 9 Y=1 Now add the two equations and solve. THEN---- Substitute your answer into either original equation and solve for the second variable. X+Y=4 X +1=4 - 1 -1 X=3 (3,1) Now check our answers in both equations-----Answer x+y=4 3+1=4 4=4 2x + 3y = 9 2(3) + 3(1) = 9 6+3=9 9=9