Solving harder linear Simultaneous Equations 7x - 4y = 37 1 2x + 3y = -6 2 In this case there is no single multiplier for one equation in order to make coefficients the same, so we have to multiply both equations Use the smallest possible multiplier to produce the smallest numbers To work with 1 x 3 21x - 12y = 111 3 2 x 4 8x + 12y = -24 4 29x = 87 4 3 + 29 x = 3 Check in 1 Substitute in 2 (2 x 3) + 3y = -6 (7 x 3) - (4 x -4) = 37 -6 3y = -12 3 y = -4

Solving harder linear Simultaneous Equations Practice 3x - 2y = 6 1 5x - 3y = 9 2 1 x 3 9x - 6y = 18 3 2 x 2 10x - 6y = 18 4 Subtract - 3 4 x = 0 Substitute in 1 (3 x 0) - 2y = 6 -2y = 6 2 y = -3 Check in 2 (5 x 0) – (3 x -3) = 9

Solving non-linear Simultaneous Equations Solve the simultaneous equations: x2 + y2 = 13 x + 3y = 3 The way to solving these is to convert them into quadratic equations in the form ax2 + bx +c Step 1 Rearrange the linear equation because this will be used to substitute into the quadratic equation. x = 3 – 3y Make x the subject of the formula because making y the subject would look like this and would be harder to substitute Step 2 Substitute x = 3 – 3y into the non-linear equation (3 - y)2 + y2 = 13 Expand and simplify (9y2 -18y + 9) + y2 = 13

Solving non-linear Simultaneous Equations Step 3 10y2 -18y + 9 = 13 Rearrange to make the equation = 0 10y2 -18y - 4 = 0 Factorise and solve the quadratic equation Step 4 (5y+1)(y-2) = 0 y = 2 Step 5 Substitute into the linear equation (easiest) When x + 3 x = 3 x = When y = 2 x + 3 x 2 = 3 x = -3 Check your answers by substitution into the non-linear equation

Solving non-linear Simultaneous Equations Practice x2 + y2 = 16 y = x - 4 Substitute into the non-linear equation x2 + (x – 4)2 = 16 Expand and simplify x2 + (x2 - 8x + 16) = 16 2x2 - 8x + 16 = 16 Rearrange to make the equation = 0 2x2 - 8x = 0 2x(x – 4) = 0 x = 0 x = 4 Substitute into the linear equation Check the values in the non linear equation When x = 0, y = -4 When x = 4, y = 0