Structures 3 Sat, 27 November 2010

11: :00 Solving simultaneous equations:  using algebra  using graphs

Simultaneous equations Equations in two unknowns have an infinite number of solution pairs. For example, x + y = 3 is true when x = 1 and y = 2 x = 3 and y = 0 x = –2 and y = 5 and so on … We can represent the set of solutions on a graph: x y x + y = 3

Simultaneous equations Another equation in two unknowns will also have an infinite number of solution pairs. For example, y – x = 1 is true when x = 1 and y = 2 x = 3 and y = 4 x = –2 and y = –1 and so on … This set of solutions can also be represented in a graph: 0 x y 3 3 y – x = 1

Simultaneous equations There is one pair of values that solves both these equations: y – x = 1 We can find the pair of values by drawing the lines x + y = 3 and y – x = 1 on the same graph. 0 x y 3 3 y – x = 1 x + y = 3 The point where the two lines intersect gives us the solution to both equations. This is the point (1, 2). At this point x = 1 and y = 2.

Simultaneous equations y – x = 1 are called a pair of simultaneous equations. x + y = 3 The values of x and y that solve both equations are x = 1 and y = 2, as we found by drawing graphs. We can check this solution by substituting these values into the original equations = 3 2 – 1 = 1 Both the equations are satisfied and so the solution is correct.

Simultaneous equations with no solutions Sometimes pairs of simultaneous equations produce graphs that are parallel. Parallel lines never meet, and so there is no point of intersection. When two simultaneous equations produce graphs which are parallel there are no solutions. How can we tell whether the graphs of two lines are parallel without drawing them? Two lines are parallel if they have the same gradient.

Simultaneous equations with no solutions We can find the gradient of the line given by a linear equation by rewriting it in the form y = mx + c. The value of the gradient is given by the value of m. Show that the simultaneous equations y – 2 x = 3 2 y = 4 x + 1 have no solutions. Rearranging these equations in the form y = mx + c gives, y = 2 x + 3 y = 2 x + ½ The gradient m is 2 for both equations and so there are no solutions.

Simultaneous equations with infinite solutions Sometimes pairs of simultaneous equations are represented by the same graph. For example, Notice that each term in the second equation is 3 times the value of the corresponding term in the first equation. 2 x + y = 3 6 x + 3 y = 9 Both equations can be rearranged to give y = –2 x + 3 When two simultaneous equations can be rearranged to give the same equation they have an infinite number of solutions.

The elimination method (1) If two equations are true for the same values, we can add or subtract them to give a third equation that is also true for the same values. For example, suppose 3 x + y = 9 5 x – y = 7

The elimination method (2) Sometimes we need to multiply one or both of the equations before we can eliminate one of the variables. For example, 4 x – y = 29 3 x + 2 y = 19 We need to have the same number in front of either the x or the y before adding or subtracting the equations.

The substitution method Two simultaneous equations can also be solved by substituting one equation into the other. For example, y = 2 x – 3 2 x + 3 y = 23

Solving problems The sum of two numbers is 56 and the difference between the two numbers is 22. Find the two numbers. The cost of theatre tickets for 4 adults and 3 children is £ The cost for 2 adults and 6 children is £44. How much does each adult and child ticket cost? In a farmyard there are sheep and chickens. How many sheep and chickens, if there are 44 heads and 112 legs.

Solving problems Remember, when using simultaneous equations to solve problems: 1) Decide what letters to use to represent each of the unknown values. 2) Use the information given in the problem to write down two equations in terms of the two unknown values. 3) Solve the simultaneous equations using the most appropriate method. 4) Check the values by substituting them back into the original problem.