Equations Mathematics

In the algebra section, we looked at how we can use letters as numbers and how to find their values. There are times, however, when life gets a little harder:

x + y = 25

We cannot say what x and y are in the equation above because there are an infinite number of possibilities:

x = 1, y = 24
x = 21, y = 4 etc.

In general, a single equation with two unknowns (we don't know x and y in this case) cannot be solved. What we need is a second equation involving at least one of the unknown variables.

Simultaneous Equations

Suppose that we were given the following equations:

x + y = 25
3x = 15

We have two different equations with two unknowns - this is enough information for us to solve them both (that is, find the values of x and y). We call these simultaneous equations because we use them at the same time.

In the second equations, we can solve immediately for x (see the algebra pages if you're not happy with this):

3x = 15 so x = 5

Now we put this value of x (called substitution) into the first equation:

5 + y = 25 so y = 20

So the only correct solution for the two equations given at the top of the page is x = 5 and y = 20.

So far so good! Let's look at a harder example:

2a + b = 21
a + b = 15

In this case we cannot just solve for a or b individually, as both variables appear in each equation. Instead we must rearrange an equation to find an unknown in terms of the other:

b = 15 - a
[This is the second equation re-stated]

Now we can substitute this statement for b into the first equation:

2a + b = 21
2a + (15 - a) = 21

This can now be tidied up, giving a correct answer for a:

2a - a = 21 - 15
a = 6

Finally we can use any of the first 3 equations to find b:

b = 15 - 6
b = 9

Testing Simultaneous Equations

Now would be a good time for you to test what you have learned. Work out the value of each variable in the examples below:

2a + b = 21 a + 2b = 24 5p + q = 100 2p - q = -30 m + n = 0 6m - n = 49

Hints: Rearrange one equation to give you one variable. Substitute it, then solve. Be careful with minus signs!

Let's see how you got on:

1. 2a + b = 21
   a + 2b = 24

   Rearrange one of the equations: b = 21 - 2a
   Next substitute in the other: a + 2(21 - 2a) = 24
   Solve: a + 42 - 4a = 24
   Rearrange: 42 - 24 = 3a
   Answer: a = 6
   Finally substitute: b = 21 - 2×6
   Answer: b = 9
   

2. 5p + q = 100
   2p - q = -30

   Rearrange one of the equations: q = 100 - 5p
   Next substitute in the other: 2p - (100 - 5p) = -30
   Solve: 2p - 100 + 5p = -30
   Rearrange: 7p = 70
   Answer: p = 10
   Finally substitute: q = 100 - 5×10
   Answer: q = 50
   

3. m + n = 0
   6m - n = 49

   Rearrange one of the equations: m = -n
   Next substitute in the other: 6(-n) - n = 49
   Solve: -7n = 49
   Answer: n = -7
   Finally substitute: m = -(-7)
   Answer: m = 7