Maths
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.
Linear and quadratic equation
Example
- Solve the simultaneous equations by drawing graphs.
- y - 2x = 1
- y = x2 - 2
Now try a Test Bite