Simultaneous Equations Calculator
Solve two linear simultaneous equations with two unknowns (x and y) step by step using the elimination method with full working shown.
Simultaneous equations calculator
Step by step calculations
Enter both equations and press Solve to see the step by step calculations.
Simultaneous Equations Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
When would you actually use this?
Two coffees and a tea cost £7.40. One coffee and three teas cost £8.20. What does each drink cost? You can't answer this with one equation alone — you need two. Simultaneous equations let you find two unknowns when you have two pieces of information about them.
Worked example
"2c + t = 7.40 and c + 3t = 8.20 — find c and t."
From eq 1: t = 7.40 − 2c
Sub into eq 2: c + 3(7.40 − 2c) = 8.20
c + 22.20 − 6c = 8.20 → −5c = −14 → c = 2.80
t = 7.40 − 5.60 = £1.80
Check in eq 2: 2.80 + 3(1.80) = 2.80 + 5.40 = 8.20 ✓
Two methods
- → Substitution: rearrange one equation for one variable, then substitute into the other. Best when one variable is already isolated.
- → Elimination: multiply equations to match coefficients, then add or subtract to remove one variable. Best for symmetrical equations.
Always verify
Substitute both answers into both original equations. One check is not enough — use both.
What are Simultaneous Equations?
Simultaneous equations are a set of two or more equations that share the same variables. The goal is to find the values for these variables that satisfy *all* the equations at the same time.
For GCSE Maths, you'll typically deal with two linear equations with two unknowns, usually 'x' and 'y'. When you solve them, you're looking for a single pair of (x, y) values that makes both equations true. Graphically, this represents the point where the two lines intersect.
How to Solve Simultaneous Equations using the Elimination Method
Make Coefficients Match
Choose one variable (either 'x' or 'y') to eliminate. Multiply one or both equations by a suitable number so that the coefficients of your chosen variable become the same (or the same number with opposite signs). Remember to multiply every term in the equation, including the constant on the right side.
Eliminate a Variable
If the coefficients of your chosen variable have the *same sign* (e.g., both +2x or both -3y), subtract one equation from the other. If they have *opposite signs* (e.g., +2x and -2x), add the equations together. This will eliminate one variable, leaving you with a single equation with one unknown.
Solve for the First Variable
Solve the new single-variable equation to find the value of that variable. This will give you either your 'x' or 'y' value.
Substitute Back
Substitute the value you just found back into one of the *original* equations (choose the simpler one!). Solve this equation to find the value of the second variable.
Check Your Solution (Optional but Recommended)
Substitute both 'x' and 'y' values into the *other* original equation to ensure they satisfy it. This confirms your solution is correct. Be aware that sometimes equations might have no solution (parallel lines) or infinitely many solutions (the same line), which you'll discover in Step 2 if both variables disappear.
Common mistakes to avoid
- Multiplying only part of an equation (e.g., 2(x + y) = 5 becomes 2x + y = 5):When you multiply an equation, you must multiply every term on both sides of the equals sign to maintain equality. 2(x + y) = 5 becomes 2x + 2y = 10.
- Sign errors when adding or subtracting equations (e.g., (2x + 3y) - (2x - y) = 4y):It's easy to forget to apply the subtraction to all terms in the second equation, especially with negative signs. (2x + 3y) - (2x - y) = 2x + 3y - 2x + y. (2x + 3y) - (2x - y) = 4y. The correct calculation is (2x - 2x) + (3y - (-y)) = 0 + 3y + y = 4y.
- Only finding one variable (e.g., finding x but forgetting to find y):Simultaneous equations require a solution for all variables. A single value isn't a complete answer. Always substitute your first variable's value back into an original equation to find the second variable.
- Substituting the found value back into the equation used to find it, rather than an original one:While it might work, it's safer to use an original equation to avoid compounding any errors made during the elimination step. After finding x, substitute it into one of the initial equations (e.g., a1x + b1y = c1) to find y.