Simultaneous Equations
Simultaneous equations are two or more algebraic equations that share variables e.g. x and y. They are called simultaneous equations because the equations are solved at the same time.
For example, below are some simultaneous equations:
2x + 4y = 14, 4x − 4y = 4
6a + b = 18, 4a + b = 14
3h + 2i = 8, 2h + 5i = −2
Each of these equations on their own could have infinite possible solutions. However when we have at least as many equations as variables we may be able to solve them using methods for solving simultaneous equations.
−3x + y = 2
When we draw the graphs of these two equations, we can see that they intersect at (1,5).
So the solutions to the simultaneous equations in this instance are:
x = 1 and y = 5
Solving simultaneous equations
When solving simultaneous equations you will need different methods depending on what sort of simultaneous equations you are dealing with.
A linear equation contains terms that are raised to a power that is no higher than one. E.g.
Linear simultaneous equations are usually solved by what’s called the elimination method (although the substitution method is also an option for you).
What is substitution?
Substitution means replacing the variables (letters) in an algebraic expression with their numerical values. We can then work out the total value of the expression.
We can substitute values into formulae to help us work out many different things. Examples range from the formula for the area of a triangle:
How to substitute a value into an algebraic expression
In order to substitute into an algebraic expression:
- Rewrite the expression substituting each letter with its given numerical value.
- Calculate the total value of the expression. Remember that you must apply BIDMAS.
Substitution is a useful method to use when one or more of the equations is not linear. Substitution works by making one of the variables the subject of one of the equations and then substituting this into the other equation.
For example, let’s solve the simultaneous equations
Step 1: Make one of the variables the subject of one of the equations.
Step 2: Substitute this into the other equation.
Step 3: Solve the equation.
Step 4: Substitute into one of the original equations to find the other value.
Step 5: Check your answer by substituting into the other original equation.
These both equal 1, so our solutions are correct.