by The Open University
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Simultaneous equations are pairs of equations that are both true (i.e. they are simultaneously true). They are both expressed as equations with two unknowns. By making one of these unknowns the subject of both equations, we can then substitute the subject in one equation and then solve for the other unknown. Then we can substitute back into the equation and solve for the subject.
We can show you what we mean with this example.
First, we need to rearrange these equations to make either x or y the subject. We have chosen to make x the subject. The order doesn't matter, as the answers that you will get must be the same.
| 2y = x + 7 | deduct 7 from both sides | 2y − 7 = x + 7 − 7 | So, 2y − 7 = x |
| y = x + 2 | deduct 2 from both sides | y − 2 = x2 − 2 | So, y − 2 = x |
Therefore, x = 2y − 7 and x = y − 2
| So, 2y − 7 = y − 2 | deduct y from both sides | 2y − y − 7 = y − y − 2 |
| So, y − 7 = − 2 | add 7 to both sides | y − 7 + 7 = − 2 + 7 |
Therefore, y = 5
We can substitute this back into either of the original equations.
2y = x + 7
So, 10 = x + 7
Therefore, x = 3
As a last check, make sure that this is true of the other equation.
x + 2 = y
x = y − 2 = 5 − 2 = 3
So the solution is x = 3, y = 5. Finally, check that these values satisfy both the original equations.
Now try to solve the following pairs of simultaneous equations for yourself.
| (a) | y = x + 10 | 3y = 2x + 5 |
| (b) | y = 4x | y = 3x + 5 |
| (c) | y = 7x + 4 | 3y = x + 7 |
| (d) | y = 2x + 4 | 3y = x + 7 |
| Equations | Step 1: rearrange to make y the subject | Step 2: solve for x | Step 3: solve for y | Solution | |
|---|---|---|---|---|---|
| (a) | y = x + 10 | y = x + 10 |  |
y = x + 10 | x = − 25 |
| 3y = 2x + 5 |  |
3x + 30 = 2x + 5 | y = − 25 + 10 = − 5 | y = − 15 | |
| x = − 25 | |||||
| (b) | y = 4x | 4x = 3x + 5 | y = 4x = 20 | x = 5 | |
| y = 3x + 5 | x = 5 | y = 20 | |||
| (c) | y = 7x + 4 | y = 7x + 4 |  |
y = 7x + 4 | x = − 0.25 |
| 3y = x + 7 |  |
21x + 12 = x + 7 | y = − 1.75 + 4 = 2.25 | y = 2.25 | |
| 20x = − 5 | |||||
| x = − 0.25 | |||||
| (d) | y = 2x + 4 | y = 2x + 4 |  |
y = 2x + 4 | x = − 1 |
| 3y = x + 7 |  |
6x + 12 = x + 7 | y = − 2 + 4 = 2 | y = 2 | |
| 5x = − 5 | |||||
| x = − 1 |
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