year 9/gcse: simultaneous equations
DESCRIPTION
Year 9/GCSE: Simultaneous Equations. Dr J Frost ([email protected]) . Last modified: 27 th August 2013. How many solutions for x and y?. For x. For y. ?. 2. ?. ∞. x 2 = 4. ?. 1. ?. ∞. x = 3. x + y = 9. ?. ∞. ?. ∞. x + y = 9 x – y = 1. ?. ?. 1. 1. 8 6 4 - PowerPoint PPT PresentationTRANSCRIPT
Year 9/GCSE: Simultaneous Equations
Dr J Frost ([email protected])
Last modified: 22nd September 2014
-10 -8 -6 -4 -2 2 4 6 8 10
8
6
4
2
-2
-4
-6
Sketch the line with equation
2x –
y = -1
Click to sketch
Starter
𝑥2=4𝑥=3
𝑥+ 𝑦=9
For For
1 ∞2 ∞∞ ∞
1 1
? ?
? ?
? ?
? ?
How many solutions for x and y?
Hint: Think about the line representing
-10 -8 -6 -4 -2 2 4 6 8 10
8
6
4
2
-2
-4
-6
By using graphical methods, solve the simultaneous equations:
2x –
y = -1
x + y = 7
Click to sketch
But why does finding the intersection of the lines give the solution?1. The line for each
equation represents all the points (x,y) for which the equation is satisfied.
2. Therefore, at the intersection(s), this gives the points for which both equations are satisfied.
?
Solution:
?
Bro Tip: To sketch a straight line, just pick two values of . If we’re sketching , say we pick , then and thus . Choose another value of and connect up.
Test Your Understanding
Copy the axis provided, and sketch the given lines on them*.Hence solve the simultaneous equations.
𝑥
𝑦 𝑦=2 𝑥−2 𝑥+ 𝑦=4
𝑥=2 , 𝑦=2
𝑥
𝑦 𝑦−𝑥=1 𝑦=−3𝑥+5
𝑥=1 , 𝑦=2
Q1 Q2
1 2 3 4 5
5
4
3
2
1
5
4
3
2
1
1 2 3 4 5
* Remember that the easiest way is to pick two points and join up, e.g. when and when .
www.wolframalpha.com
Thinking graphically…For two simultaneous equations, when would we have…
0 solutions for and ? Lines are parallel but not the same.
Infinitely many solutions for and ?
Lines are the same.e.g.
?
?
Exercise 1For each of the following, sketch axis for from 0 to 6 and from 0 to 6. Sketch the two lines on your axis and use them to estimate the solution to the simultaneous equations.
Consider the simultaneous equations:
where and are constants. By thinking about the lines corresponding to the equations, under what conditions will we have:a) Infinitely many solutions for and ?
b) No solutions:
For each of the following, sketch axis for from -5 to 5 and from -5 to 5. Sketch the two lines on your axis and use them to estimate the solution to the simultaneous equations.
Given that:
Sketch suitable lines to estimate the solutions to these simultaneous equations.
1
2
3
a
b
c
a
b
c
N
?
?
?
??
?
?
?
?
Three methods of solving simultaneous equations
graphicallyby
elimination
by substitution
By either adding or subtracting the equations, we can ‘eliminate’ one of the variables.
METHOD #2: Solving by Elimination
2 𝑥+𝑦=6 1
2
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.21 +
Obtain by substituting your known into one of the two equations.
4 𝑥+ 𝑦=6 1
2
12 - 2 𝑥=−2
?
?
You can solve in 2 different ways:•Eliminating .•Eliminating .
𝑥=3 , 𝑦=−1?
Solving by Elimination
1+2?
Test Your Understanding
𝑥=4 , 𝑦=2
𝑥=−1 , 𝑦=5
𝑥=9 , 𝑦=−4
?
?
?
If you finish quickly: 1+21+32+3
?
Exercise 2
If using textbook: GCSE Rayner (Old Edition) Page 105 - Exercise 28
Solve the following by substitution.
6
Solve:
[Cayley] Mars, his wife Venus and grandson Pluto have a combined age of 192. The ages of Mars and Pluto together total 30 years more than Venus’ age. The ages of Venus and Pluto together total 4 years more than Mars’s age. Find their three ages.Hint: You can form 3 equations with 3 unknownsMars = 94, Venus = 81, Pluto = 17
[Maclaurin] Find all integer values that satisfy the following equations:
Adding:
When When
Thus
Two cats and a dog cost £91. Three cats and two dogs cost £159. How much does a cat cost?£23
1
a
b
c
d
2 a
b
c
4
N1
N2
3
?
?
?
?
?
?
?
?
?
?
?
(1,7)
(3,175)
The graph shows two points (1,7) and (3,175) on a line with equation:
Determine and (where and are positive constants).
Answer:a = 5, k = 1.4?
Harder GCSE Exam Question
by substitution
Three methods of solving simultaneous equations
graphicallyby
elimination
We currently have two equations both involving two variables.
Perhaps we could put one equation in terms of or , then substitute this expression into the other.
2x + y = 7 y = 7 – 2x3x – 2y = 03x – 2(7-2x) = 03x – 14 + 4x = 07x – 14 = 07x = 14x = 2Then y = 3
?
METHOD #3: Solving by Substitution
Why do you think we chose this equation to rearrange??
Answer:x = 2, y = 1 ?
Your go…
Answer:?
Solve for and , using substitution.
Exercise 3Use substitution only to solve the following simultaneous equations.
[Cayley] James, Alison and Vivek go into a shop to buy some sweets. James spends £1 on four Fudge Bars, a Sparkle and a Chomper. Alison spends 70p on three Chompers, two Fudge Bars and a Sparkle. Vivek spends 50p on two Sparkles and a Fudge Bar. What is the cost of a Sparkle?Sparkle = 15p
[Maclaurin] Solve the simultaneous equations:
(You must have proved algebraically, using substitution, that these are the only solutions)
[Maclaurin] Solve:
(Hint: If after substitution you end up with a cubic equation, you can sometimes factorise it by factorising the first two terms and the last two terms first separately)
𝑥
𝑥3 𝑦
A
B
C
The angle at is 12° greater than the angle at . Find and .
Magnus wants to buy 80 Ferraris, some yellow and some red. He must spend the whole of the £20m of his weekly pocket money. He buys yellow Ferraris at £40k and red Ferraris at £320k. How many Ferraris of each type did he buy?
1 2
3
5
N1
N2
?
?
?
?
?
?
?
?
?
?£13 £19 £17
4 What is the cost of a cat?£1
a
b
c
d
e
?
by substitution
Three methods of solving simultaneous equations
graphicallyby
elimination
SECRET LEVEL
by matrices
