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) □ Its radius is r

Consider x 2 + y 2 = 9 4 -4 x -3 -2 3 x When yx == 0, 0, 2 xy 22 == 99 1 So So xy == +3 +3 or or -3 -3 -1 1 -1 -2 -3 x -4 2 x 3 4 We have 2 points x (0, -3) (3, 0) (0, 3) and (-3, 0)

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

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

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

…. Once drawn 4 1 st Solution x = 0. 94 y = 2. 85 3 2 2 nd Solution x = -1. 75 y = -2. 41 1 -4 -3 -2 -1 1 -1 -2 -3 -4 2 3 4 x

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

Use formula 5 x 2 + 4 x -8 = 0 So, x = -1. 7266…. or 0. 9266…. So, y = -2. 453…. 2. 8532…. or Solutions, (-1. 73, -2. 45) & (0. 93, 2. 85) to 2 d. p.