Simultaneous Equations
Simultaneous equations means you must solve at least two equations at the same time
Normally you will have to solve just 2 equations
3x – 2y = 16
5x + 4y = 12
REMEMBER
You are trying to find the value of x and the value of y that makes BOTH equations true
Aims for this topic:
You will know what simultaneous equations are
You will be able to solve simultaneous equations using graphs
You will be able to solve simultaneous equations using an algebraic method
You will be Learn how to solve real problems by forming and solving simultaneous equations using an algebraic method
On completion of this unit a learner should:
Know the basic underpinning mathematical techniques- methods used to manipulate and/or solve formulae, equations and algebraic expressions
PLAY IT!
There are FIVE stages to solving simultaneous equations
1. EQUATE
2. ELIMINATE
3. SOLVE
4. SUBSTITUTE
5. CHECK
5
5 5 5 5 5 55 33333 55555555555
3333333333333
33
-
-22
-22
-22
-22
-22
-22
44 4
4444
444
44
y =
-22
-22
STAGE 1: EQUATE x
3x – 2y = 16 ……….. (1)
5x + 4y = 12 ………….. (2)
Step 1: Multiply equation 1 by the Coefficient of x in equation 2 15x – 10y = 80 ……….. (3)
15x + 12y = 36 ………….. (4)
Step 2: Multiply equation 2 by the Coefficient of x in equation 1
STAGE 2: Eliminate
15x – 10y = 80 ……….. (3) 15x + 12y = 36 ……….. (4)
PLAY IT!
– 22y = 44 ……….. (5)
We must eliminate the terms containing x by adding or subtracting equation 3 and 4
STAGE 3: Solve
We have to solve equation 5 to find the value of y
-22y = 44
We do this as follows
Therefore y = -2
REMEMBERy = -2 for both equations
-2 -2 -2-2-2-2-2-2
-2-2
-2-2-2-2
USING GRAPHS
STAGE 4: Substitute
We know that y = -2 (for both equations)
Equation 1 is3x -2y = 16
PLAY IT! So to substitute We re-write equations 1 as
3 (x) -2 (-2) = 16
This gives the equation REMEMBER 3x + 4 = 16 Which gives x = 4 x = 4 for both equations
STAGE 5: Check
We know that x = 4 and y = -2
We can check this result by substituting these values in equation 2
Equation 2 5x + 4y = 12
Substituting gives5(4) + 4(-2) = 12
5 x 4 = 20 4 x –2 = -8
So 20 – 8 does equal 12
Answer is x = 4, y = -2
There are just three easy steps: Do a table of values for each equation Draw the two graphs Write down the x and y values where the graphs cross
Solving simultaneous equations using graphs
Draw the two graphs:
0
1
2
3
4
5
6
7
-3 -2 -1 0 1 2 3
x
y
y = 3 - x x + 2y = 4
Here is an example:Solve simultaneously
y = 3 - xx + 2y = 4
y = 3 -x
X -3 -2
-1
0
1 2
3
y 6 5 4 3
2 1
0
x + 2y = 4
X -3 -2
-1 0 1 2 3
y 3.5 3 2.5 2 1.5 1 0.5
Write down the x and y values where the
graphs cross :
0
1
2
3
4
5
6
7
-3 -2 -1 0 1 2 3
x
y
y = 3 - x x + 2y = 4