Simultaneous Equations

From: Dominic Walsh
Date: 2 September 1999

Can you give me some guidance how to solve the simultaneous equations:
2x + 4y = 10 and
3x - y = -6
using substitution.

The substitution method works as follows:

Rearrange one equation to make one of the variables the subject;
Substitute this expression in the other equation;
Solve the resulting equation;
Use your solution to this variable to find the value of the other variable, by substituting back into one of the original equations.

In your case, it is easiest to make y the subject of the second equation:

y = 3x + 6

Now use this expression for y in your first equation:

2x + 4(3x + 6) = 10

Solve this equation for x:

2x + 12x + 24 = 10

14x + 24 = 10

14x = -14

x = -1

Now use your expression for y to finish the solution:

y = 3x + 6

y = 3(-1) + 6

y = -3 + 6

y = 3

So the solution to the simultaneous equations is:

x = -1
y = 3