Simultaneous equations – Higher
Sometimes you will be asked to find 2 unknown values by solving 2 equations at the same time. These types of equations are called simultaneous equations.
Simultaneous equations
Simultaneous equations are two equations with two unknowns. They are called simultaneous because they must both be solved at the same time.
The first step is to try to eliminate one of the unknowns.
Example
Solve these simultaneous equations and find the values of x and y.
* Equation 1: 2x + y = 7
* Equation 2: 3x – y = 8
Add the two equations to eliminate the ys:
* 2x + y = 7
* 3x – y = 8
* ————
* 5x = 15
* x = 3
* Now you can put x = 3 in either of the equations.
* Substitute x = 3 into the equation 2x + y = 7:
* 6 + y = 7
* y = 1
So the answers are x = 3 and y = 1
Sometimes you will need to multiply one of the equations before you can add or subtract. Have a look at the activity below.
On other occasions you will need to multiply both equations to find the unknown values. Always look for a way to cancel out one of the unknown terms. Have a go at the question below.
Question
* Solve the two equations:
* Equation 1: 2a – 5b = 11
* Equation 2: 3a + 2b = 7
Answer
* 4a – 10b = 22 (Multiply by 2)
* 15a + 10b = 35 (Multiply by 5)
* ———————-
* 19a = 57 (Adding)
* a = 3
* Put a = 3 into the equation 3a + 2b = 7:
* 9 + 2b = 7
* 2b = -2
* b = -1
So the answers are a = 3 and b = -1.
We can also use a method of substitution. Look at the following example:
Example
Solve the simultaneous equations:
Equation 1: y – 2x = 1
Equation 2: 2y – 3x = 5
Rearranging Equation 1, we get y = 1 + 2x
We can replace the ‘y’ in equation 2 by substituting it with 1 + 2x
Equation 2 becomes: 2(1 + 2x) – 3x = 5
2 + 4x – 3x = 5
2 + x = 5
x = 3
Substituting x = 3 into Equation 1 gives us y – 6 = 1, so y = 7.
Solving simultaneous equations using graphs
Solving simultaneous equations using a graph is easier than you might think. First, you need to draw the lines of the equations. The points where the lines cross is the solution.
Linear equations
The graphs of linear equations will give straight lines.
Example
* Solve these simultaneous equations by drawing graphs:
* 2x + 3y = 6
* 4x – 6y = – 4
For example, to draw the line 2x + 3y = 6 pick two easy numbers to plot. One when x = 0 and one where y= 0
* When x = 0 in the equation 2x + 3y = 6
* This means 3y = 6 so y = 2
* So one point on the line is (0, 2)
* When y = 0
* 2x = 6 so x = 3
* So another point on the line is (3 ,0)
In an exam, only use this method if you are prompted to by a question. It is usually quicker to use algebra if you are not asked to use graphs.
image: graph showing 2x + 3y = 6 and 4x minus 6y = minus 4
Linear and quadratic equation
Example
* Solve the simultaneous equations by drawing graphs.
* y – 2x = 1
* y = x2 – 2
Simultaneous Equations
A pair of “Simultaneous equations” is two equations which are both true at the same time. You have two equations which have two unknowns to be found.
Example
A man buys 3 fish and 2 chips for £2.80
A woman buys 1 fish and 4 chips for £2.60
How much are the fish and how much are the chips?
First we form the equations. Let fish be f and chips be c.
We know that:
3f + 2c = 280 (1)
f + 4c = 260 (2)
There are two methods of solving simultaneous equations. Use the method which you prefer:
Elimination
This involves changing the two equations so that one can be added/ subtracted from the other to leave us with an equation with only one unknown (which we can solve). We can ‘change’ the equations by multiplying them through by a constant- as long as we multiply both sides of the equation by the same number it will remain true.
In our above example:
Doubling (1) gives:
6f + 4c = 560 (3)
Since equation (2) has a 4c in it, we can subtract this from the new equation (3) and the c’s will all have disappeared:
(3)-(2) gives 5f = 300
∴ f = 60
Therefore the price of fish is 60p
So we can put f=60 in either of our original equations. Substitute this value into (1):
3(60) + 2c = 280
∴ 2c = 100
c = 50
Therefore the price of chips is 50p
Substitution
The method of substitution involves transforming one equation into x = (something) or y = (something) and then substituting this something into the other equation.
So,
Rearrange one of the original equations to isolate a variable.
Rearranging (2): f = 260 – 4c
Substitute this into the other equation:
3(260 – 4c) + 2c = 280
∴ 780 – 12c + 2c = 280
∴ 10c = 500
∴ c = 50
Substitute this into one of the original equations to get f = 60 .
This section is higher tier Harder simultaneous equations
To solve a pair of equations, one of which contains x2, y2 or xy, we need to use the method of substitution.
Example
2xy + y = 10 (1)
x + y = 4 (2)
Take the simpler equation and get y = …. or x = ….
from (2), y = 4 – x (3)
this can be substituted in the first equation. Since y = 4 – x, where there is a y in the first equation, it can be replaced by 4 – x .
sub (3) in (1), 2x(4 – x) + (4 – x) = 10
∴ 8x – 2×2 + 4 – x – 10 = 0
∴ 7x – 2×2 – 6 = 0
∴ 2×2 – 7x + 6 = 0 (taking everything to the other side of the equals sign)
∴ (2x – 3)(x – 2) = 0
∴ either 2x – 3 = 0 or x – 2 = 0
therefore x = 1.5 or 2 .
Substitute these x values into one of the original equations.
When x = 1.5, y = 2.5
when x = 2, y = 2
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