Make one of the unknowns the subject of the linear equation (rearranging where necessary) Use the linear equation to substitute into the quadratic equation There are usually two pairs of solutions equation 2: 22 + 32 = 13 YES and (−3)2 + (−2)2 = 13 YES 1 Substitute x + 1 for y into the second equation
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[PDF] Solving linear and quadratic simultaneous equations
Make one of the unknowns the subject of the linear equation (rearranging where necessary) Use the linear equation to substitute into the quadratic equation There are usually two pairs of solutions equation 2: 22 + 32 = 13 YES and (−3)2 + (−2)2 = 13 YES 1 Substitute x + 1 for y into the second equation
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(d) y = 2x² + 9x + 1 (e) y = 2x² + x + 1 (f) y = −x² + 5x + 2 y = 3x + 9 y = x² − 5x − 7 y = 3x² − x − 2 Question 2: Solve the following simultaneous equations
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Scheme of work: 1c. Equations quadratic/linear simultaneousKey points
Make one of the unknowns the subject of the linear equation (rearranging where necessary). Use the linear equation to substitute into the quadratic equation.There are usually two pairs of solutions.
Examples
Example 1 Solve the simultaneous equations y = x + 1 and x2 + y2 = 13 x2 + (x + 1)2 = 13 x2 + x2 + x + x + 1 = 132x2 + 2x + 1 = 13
2x2 + 2x 12 = 0
(2x 4)(x + 3) = 0So x = 2 or x = 3
Using y = x + 1
When x = 2, y = 2 + 1 = 3
When x = 3, y = 3 + 1 = 2
So the solutions are
x = 2, y = 3 and x = 3, y = 2Check:
equation 1: 3 = 2 + 1 YES and 2 = 3 + 1 YES equation 2: 22 + 32 = 13 YES and (3)2 + (2)2 = 13 YES1 Substitute x + 1 for y into the second
equation.2 Expand the brackets and simplify.
3 Factorise the quadratic equation.
4 Work out the values of x.
5 To find the value of y, substitute
both values of x into one of the original equations.6 Substitute both pairs of values of x
and y into both equations to check your answers. Example 2 Solve 2x + 3y = 5 and 2y2 + xy = 12 simultaneously. 532 yx