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Solve Simultaneous Equations One Linear, one quadratic [Circle]

Solve Simultaneous Equations One Linear, one quadratic [Circle]. GCSE Higher. Content. Equation of a circle Equation of straight line Graphical Solution Algebraic Solution [Substitution Method]. Equation of a Circle. Is x 2 + y 2 = r 2 Where The circle has centre (0,0)

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Solve Simultaneous Equations One Linear, one quadratic [Circle]

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  1. Solve Simultaneous EquationsOne Linear, one quadratic [Circle] GCSE Higher

  2. Content • Equation of a circle • Equation of straight line • Graphical Solution • Algebraic Solution[Substitution Method]

  3. Equation of a Circle • Is x2 + y2 = r2 • Where • The circle has centre (0,0) • Its radius is r

  4. 4 3 2 1 -1 -2 -3 -4 -4 -3 -2 -1 1 2 3 4 x Consider x2 + y2 = 9 When y = 0, x2 = 9 So x = +3 or -3 When x = 0, y2 = 9 So y = +3 or -3 x x x We have 2 points (3,0) We have 2 points (0,3) and (0,-3) and (-3,0) x

  5. Equation Straight Line • y = mx + c • Where • m is the gradient or slope • c is the y-intercept

  6. 4 3 2 1 -1 -2 -3 -4 -4 -3 -2 -1 1 2 3 4 Consider y = 2x + 1 y intercept Point (0,1) x Gradient = 2 Line rises 2 units for every 1 unit to the right

  7. Solve these 2 equations simultaneously • Graphical method • May be required to draw one or both equations • Careful drawing required for accurate answer

  8. 4 3 2 1 -1 -2 -3 -4 -4 -3 -2 -1 1 2 3 4 x …. Once drawn 1st Solution x = 0.94 y = 2.85 2nd Solution x = -1.75 y = -2.41

  9. Algebraic Solution y = 2x + 1 x2 + y2 = 9 Substitute 2x + 1 for y x2 + (2x + 1)2 = 9 Expand (2x + 1)2 x2 + 4x2 + 4x + 1 = 9 Simplify 5x2 + 4x + 1 = 9 Rearrange 5x2 + 4x -8 = 0

  10. Use formula 5x2 + 4x -8 = 0 So, x = -1.7266…. or 0.9266…. So, y = -2.453…. or 2.8532…. Solutions, (-1.73, -2.45) & (0.93,2.85) to 2d.p.

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