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Solving Systems of Equations

Solving Systems of Equations. f(x)= -1/2x 2 + 18. f(x)= 2x 2 - 2. by Fen Xu and Timothy Lou Ly. The Concept. y = 1x. Graphing a Linear Equation. (12, 20). (10, 16). (8, 12). y = 2x - 4. (-8, -8). (-16, -16). (-20, -20). The Concept. Dependent System. Inconsistent System.

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Solving Systems of Equations

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  1. SolvingSystems of Equations f(x)= -1/2x2 + 18 f(x)= 2x2 - 2 • by Fen Xu and Timothy Lou Ly

  2. The Concept y = 1x Graphing a Linear Equation (12, 20) (10, 16) (8, 12) y = 2x - 4 (-8, -8) (-16, -16) (-20, -20)

  3. The Concept Dependent System Inconsistent System Two overlapping lines with the same slope and points Two lines with the same slopes that never intersect or share points.

  4. The Concept Three Ways to Solve - Graphing - Addition - Substitution

  5. _ _ x - 1 y + 2 + = 4 2 3 Addition simplest form Step 1: Make equations into simplest form of Ax + By = C. Ax + By = C x - 2y = 5 6 ( ) Multiply by LCD to get rid of fractions = 3x - 3 + 2y + 4 = 24 - 3 + 4 Combine like terms by adding -3 and +4 together = 3x + 2y - 1 = 24 24 Add +1 to both sides to cancel -1 and isolate variables = 3x + 2y = 23 solve for “x” or “y.” Step 2: Choose to solve for “x” or “y.” For the example, we’ll solve for “x.”

  6. _ Addition Step 3: Add the two equations so that the y-values cancel. Add equations so y-values cancel. / 3x + 2y = 23 Because 2y is being subtracted from 2y, they cancel / + ( x - 2y = 5) If you wanted to solve for “y” and cancel “x,” you would need to multiply the 2nd equation by -3 4x = 28 solve for “x.” Step 4: Continue to solve for “x.” 4 4x = 28 Divide both sides by 4 x = 7

  7. Addition Step 5: We can now put 7 in for “x” in any equation and find the value of “y.” put 7 in for “x” find “y.” x - 2y = 5 CHECK: x- 2y = 5 3x + 2y = 23 3(7) + 2(1) = 23 (7) - 2(1) = 5 = 7 - 2y = 5 -2 = -2y = -2 (7) - 2 = 5 21 + 2 = 23 √ √ 23 = 23 = y = 1 5 = 5 That’s it! (7, 1) is your solution/intersection!

  8. _ _ x - 1 y + 2 + = 4 2 3 _ -3 x 11.5 =y = + 2 Substitution Solve for one variables one equation Step 1: Solve for one of the variables from one equation. x - 2y = 5 6 ( ) Multiply by LCD to get rid of fractions = 3x - 3 + 2y + 4 = 24 - 3 + 4 Combine like terms by adding -3 and +4 together = 3x + 2y - 1 = 24 24 Add +1 to both sides to cancel -1 and isolate variables = 3x + 2y = 23 3x Subtract 3x from both sides = 2y = -3x + 23 2 Divide the equation by 2

  9. _ _ -3 -3 x x 11.5 11.5 y = + + 2 2 Substitution Step 2: Substitute “ ” for “y” in the other equation: x - 2y = 5 and solve. - 2( ) ) = 5 x - 2( Multiply out -2 - 23 5 = x + 3x - 23 = 5 Add 23 to both sides 4 = 4x = 28 Divide the equation by 4 =x = 7 7 for “x,” y = 1 Step 3: Putting in 7 for “x,” we know that y = 1. -2 x - 2y = 5 = 7 - 2y = 5 = -2y = -2 = y = 1

  10. _ -3 x y = + 11.5 2 _ 1 x - 2.5 y = 2 (0, 11.5) (4, 5.5) (4, -0.5) (0, -2.5)

  11. Review Try to solve this with the method of your choice: 7x-6y=-6 -7x+6y=-4 We’ll check, and if you get it right, you get some candy! (You’ll all get some anyways)

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