Non - Graphical Solution of Simultaneous Equations Equation: x + 5 = 10 We can solve easily – one equation, one variable. Can we find a single solution to this equation? y = 3x + 2 No, because with 2 variables it states the equation of a straight line A linear equation In order to solve for 2 variables we are finding where two lines intersect.

Non - Graphical Solution of Simultaneous Equations Solve this pair of simultaneous equations: x + 2y = 7 and x = y - 8 First, write one equation directly under the other, keeping the ‘=‘ Vertically in line and number the equations x + 2y = 7 1 x = y - 8 2 Rearrange the equations if necessary so that both unknown variables are on the same side of the equation. If the equation looks different give it a new number x + 2y = 7 1 x – y = - 8 3

Non - Graphical Solution of Simultaneous Equations x + 2y = 7 1 x – y = - 8 3 We can now use the 2 equations together If we subtract equation 3 from equation 1 1 3 - 3y = 15 y = 5 Now we have found one variable we can use this by substitution into one of the original equations to find the other variable Substitute in 1 x + 2 x 5 = 7 x + 10 = 7 x = -3 We can check to see whether we have the right answers by substituting the values in the other original equation. -3 = 5 - 8

Non - Graphical Solution of Simultaneous Equations Practice 3x – y = 10 1 x + y = -2 2 1 2 + 4x = 8 x = 2 Substitute in 2 2 + y = -2 y = -4 Check in 1 (3 x 2) – (-4) = 10

Non - Graphical Solution of Simultaneous Equations What happens when there is not a common coefficient of the a variable between the two equations? 5x + 3y = 27 1 2 2x + y = 10 because there is not a common coefficient we need to make one by multiplying one of the equations. Always look for the easiest way Multiplying by 3 would be easiest because then we would have 3y 2 3 6x + 3y = 30 We could subtract from But that would leave us with negative numbers…. Messy 1 3 Now subtract from . 3 6x + 3y = 30 5x + 3y = 27 1 Check in 1 3 1 - x = 3 5 x 3 + 3 x 4 = 27 1 Substitute in 2 2 x 3 + y = 10 y = 4

Non - Graphical Solution of Simultaneous Equations Practice 6x + 2y = -10 1 2x + 4y = -10 2 1 x 2 12x + 4y = -20 3 Subtract - 2 3 12x + 4y = -20 3 2x + 4y = -10 2 2 3 - 10x = -10 x = -1 Substitute in 1 Check in 2 (6 x -1) + 2y = -10 (2 x -1) + (4 x -2) = -10 -6 + 2y = -10 2y = -4 y = -2