What Are Simultaneous Equations?
Imagine you're at a food stall. You know that 2 burgers and 1 drink cost 7. Can you figure out the price of each item? This is exactly the kind of problem simultaneous equations help us solve.
Two or more equations that share the same unknown variables and are both true at the same time. The solution is the set of values that satisfies all the equations simultaneously.
For the burger example, we can write:
where = price of a burger and = price of a drink.
We need to find values of and that make both equations true at the same time. We'll find that a burger costs 3 — let's see how to get there using two powerful methods: substitution and elimination.
Method 1: Solving by Substitution
The substitution method works by rearranging one equation to express one variable in terms of the other, then substituting that expression into the second equation.
A technique for solving simultaneous equations where you isolate one variable in one equation and then replace (substitute) it into the other equation to solve for the remaining variable.
Step-by-step process:
- Choose one equation and rearrange it to make one variable the subject (e.g., )
- Substitute this expression into the other equation
- Solve the resulting equation (it will now have only one variable)
- Back-substitute the value you found into the rearranged equation to find the other variable
- Check your answer in both original equations
Choose the equation that's easiest to rearrange — look for a variable that already has a coefficient of 1 or −1. This avoids messy fractions!
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