Year 9 Simultaneous Equations Dr J Frost jfrosttiffin

Year 9: Simultaneous Equations Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 13 th September 2015

How many solutions for x and y? 1? 2? ? ? 1?

8 6 ? 2 x – y = -1 7 = y x + 4 But why does finding the intersection of the lines give the solution? 2 1. The line for each equation represents all the points (x, y) for which the equation is satisfied. -10 -8 -6 -4 -2 2 4 6 8 10 2. Therefore, at the ? intersection(s), this gives the points for which both -2 equations are satisfied. -4 -6 Click to sketch

Test Your Understanding Copy the axis provided, and sketch the given lines on them*. Hence solve the simultaneous equations. Q 1 Q 2 5 5 4 4 3 3 2 2 1 2 3 4 5

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Thinking graphically… For two simultaneous equations, when would we have… Lines are parallel but not the same. ?

Exercise 1 1 3 a a ? ? b c b ? ? c ? ? 2 N ? ? ?

Three methods of solving simultaneous equations by substitution graphically by elimination

METHOD #2: Solving by Elimination By either adding or subtracting the equations, we can ‘eliminate’ one of the variables. 1 2 1 + 2 ? 1 2 2 ? - 1 Bro Tip: I strongly urge you to number your equations. This becomes crucial when you have three equations/three unknowns, so that you can indicate which equations you are combining.

Solving by Elimination ? ?

Test Your Understanding ? ? ?

Elimination by other means ?

Exercise 2 1 ? a ? b ? ? b N 2 c ? ? c ? 4 ? N 3 ? d 5 ? 2 N 1 3 a [IMC 2004 Q 5] The sum of two numbers is 2. The difference between them is 4. What is their product? -3 ? 6 ? ? ?

Three methods of solving simultaneous equations by substitution graphically by elimination

METHOD #3: Solving by Substitution We currently have two equations both involving two variables. ? ? Why do you think we chose this equation to rearrange?

Check Your Understanding Answer: x = 2, ? y = 1 ?

Exercise 3 Use substitution only to solve the following simultaneous equations. A 1 a 2 5 ? b C B ? c ? ? ? N 1 3 ? d ? e ? ? 4 N 2 £ 13 £ 19 £ 17 What is the cost of a cat? £ 1 ? ?

Three methods of solving simultaneous equations (you’ll have to wait for Further Maths A Level for this one) by substitution graphically by elimination SECRET LEVEL by matrices