Download Solve_Simultaneous_Equations_Linear___Quadratic

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 x2 + y2 = r2
□ Where
□ The circle has centre (0,0)
□ Its radius is r
Consider x2 + y2 = 9
4
-4
x
-3
-2
3 x
When
When yx == 0,
0,
2
xy22 == 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 = 2x + 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
1st Solution
x = 0.94
y = 2.85
3
2
2nd Solution
x = -1.75
y = -2.41
1
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
x
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
Use formula
5x2
+ 4x -8 = 0
b  b2  4ac
x
2a
4  42  4 x5x  8
x
2 x5
4  176
x
10
So, x = -1.7266…. or
0.9266….
So, y = -2.453….
2.8532….
or
Solutions, (-1.73, -2.45) & (0.93,2.85)
to 2d.p.