Free Simultaneous Equations Solver
Solve systems of linear equations step-by-step. Learn elimination, substitution, and matrix methods with instant feedback. Perfect for GCSE and A-Level algebra.
Step-by-step learning with explanations
What are Simultaneous Equations?
Simultaneous equations (also known as a system of equations) are a set of two or more equations that contain the same variables. The goal is to find the specific values for the variables (like x and y) that make every equation in the set true at the same time.
The General Concept
1. Multiple Unknowns
You usually have as many equations as you have unknowns (e.g., 2 equations for x and y).
2. Shared Solution
The values you find must work in both Equation A and Equation B perfectly.
3. Geometric Intersection
On a graph, the solution is the exact point where the lines or planes cross each other.
4. Real-World Logic
Used to find break-even points, mixture concentrations, and relative speeds.
Solving Methods & Techniques
There are three primary ways to solve linear systems. Our solver supports all of them with full working:
Elimination Method
The "standard" algebraic approach. You multiply one or both equations by a factor so that the coefficients of one variable match, then add or subtract the equations to cancel that variable out.
Same Sign Subtract (SSS)
Standard form: ax + by = c
Medium (GCSE Core)
Substitution Method
Rearrange one equation to isolate a single variable (e.g., y = 2x - 3). Then, replace that variable in the second equation with the expression you just found.
Substitute using brackets
When x or y has coefficient 1
Easy to Moderate
Matrix Method (Cramer's Rule)
A systematic way to solve systems using linear algebra. By calculating the determinants of specific matrices, you can find the values of x, y, and z directly without rearranging.
x = Dx / D
3x3 systems or larger
A-Level / Higher
Substitution vs. Elimination
Choosing the right method can save you minutes in an exam. Here is how to decide:
| Scenario | Recommended Method | Why? |
|---|---|---|
| One variable is already isolated (y = 3x) | Substitution | Direct replacement is fastest |
| Both equations look like 2x + 3y = 10 | Elimination | Easier to match coefficients |
| Variable has coefficient of 1 (x + 4y = 9) | Substitution | Easy to rearrange without fractions |
| Coefficients are multiples (2x and 4x) | Elimination | Only one equation needs multiplying |
| Messy decimals or large numbers | Elimination | Less prone to algebraic expansion errors |
Possible Outcomes of a System
Not every set of simultaneous equations has a single answer. Here are the three cases:
| Outcome | Algebraic Result | Geometric Meaning |
|---|---|---|
| One Solution | x = a, y = b | Lines intersect at one point |
| No Solution | Impossible (e.g., 0 = 5) | Lines are parallel |
| Infinite Solutions | Identity (e.g., 0 = 0) | Lines are the same |
How the Solver Works
Input Equations
Type your equations using the scientific keyboard. We handle fractions and decimals.
Analyze System
The tool detects if it is 2x2 or 3x3 and picks the most efficient path.
Step-by-Step Working
We show multipliers, addition/subtraction, and substitution steps clearly.
Verify & Check
The solution is plugged back into the original equations to prove it is correct.
Common Mistakes in Systems
Avoid these frequent errors to ensure you get full marks in your algebra exams:
Adding when you should Subtract
If the coefficients you are eliminating have the same sign (both positive), you must subtract the equations. Adding them will just double the variable instead of removing it.
Remember SSS: Same Sign Subtract. Only add if the signs are different (+3y and -3y).
Forgetting to multiply the constant
When multiplying an entire equation by a number (like 2), many students forget to multiply the number on the right side of the equals sign.
Imagine the multiplier applies to every single term in the row, including the constant.
Incorrect substitution with negative numbers
When substituting x = -2 into 3x + y = 10, it is easy to write 6 + y instead of -6 + y.
Always wrap negative numbers in brackets during substitution: 3(-2) + y = 10.
Only finding the first variable
Simultaneous means two or more. If you find x, you are only halfway there. Many students lose 2 marks by stopping early.
Immediately substitute your first answer back into an original equation to find the second value.
Subtracting negatives incorrectly
In elimination, subtracting a negative (e.g., 5y - (-2y)) becomes addition (7y). This is the #1 source of errors.
Write out the subtraction explicitly on the side: 5 - (-2) = 7.
Not checking the answer
Algebra errors are easy to make. A solution might work in Equation 1 but fail in Equation 2.
Plug your final (x, y) into BOTH equations. If both sides match, you are 100% correct.
Worked Examples
Practice with these GCSE and A-Level style problems:
Example 1: Solve 3x + y = 11 and x + y = 5
Solution:
Step 1: Subtract Equation 2 from Equation 1.
(3x - x) + (y - y) = 11 - 5
2x = 6 → x = 3
Step 2: Substitute x = 3 into Equation 2.
3 + y = 5 → y = 2
Check: 3(3) + 2 = 11. Correct.
Example 2: Solve 2x + 3y = 12 and 5x + 2y = 19
Solution:
Step 1: Multiply Eq 1 by 2 and Eq 2 by 3 to match y-coefficients.
(1) 4x + 6y = 24
(2) 15x + 6y = 57
Step 2: Subtract (1) from (2).
11x = 33 → x = 3
Step 3: Substitute x = 3 into Eq 1.
2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2
Example 3: Solve y = 2x - 1 and 3x + 2y = 19
Solution:
Step 1: Substitute (2x - 1) for y in the second equation.
3x + 2(2x - 1) = 19
3x + 4x - 2 = 19 → 7x = 21 → x = 3
Step 2: Substitute x = 3 into the first equation.
y = 2(3) - 1 = 5 → y = 5
Example 4: Solve x+y+z=6, 2y+5z=-4, 2x+5y-z=27
Quick Logic:
1. Use Cramer's Rule or Gaussian Elimination.
2. Find Determinant D of coefficients = -21.
3. Solve for x, y, z using determinants Dx, Dy, Dz.
4. Solutions: x=5, y=3, z=-2.
Exam Tips for Simultaneous Equations
Always label your equations
Call them (1) and (2). This makes your working much easier for the examiner to follow (e.g., "Multiply (1) by 5").
Double check with the OTHER equation
If you used equation (1) to find y, use equation (2) to check your final x and y values. They must work for both.
Look for the path of least resistance
If an equation has a "y" on its own, use substitution. If both have "3x" or "5y", use elimination immediately.
Keep your work lined up
Keep your x terms, y terms, and equals signs in vertical columns. This prevents simple alignment errors.
Don't be afraid of fractions
Exam answers are often integers (like 2 or -5), but they can be fractions like 1/2 or 3/4. Don't assume you're wrong if you get a decimal.
Check for non-linear systems
If one equation has an x², you must use substitution. Elimination only works for linear systems (ax + by = c).
Frequently Asked Questions
How do you solve simultaneous equations?
There are three main methods: 1. Elimination: multiply equations so one variable has the same coefficient, then add/subtract. 2. Substitution: rearrange one equation to isolate a variable and substitute it into the other. 3. Graphical: plot both lines and find where they cross.
What is the elimination method?
The elimination method involves multiplying one or both equations by constants so that the coefficients of one variable (e.g., x) are equal. You then add or subtract the equations to "eliminate" that variable, leaving an equation with only one unknown to solve.
When should I use substitution instead of elimination?
Substitution is best when one equation is already rearranged (like y = 2x + 1) or when a variable has a coefficient of 1 (like x + 3y = 7), making it easy to isolate. Elimination is usually faster for equations in the form ax + by = c.
How do you solve simultaneous equations with 3 variables?
To solve a 3x3 system (x, y, z), use elimination to reduce it to a 2x2 system. Pick two pairs of equations and eliminate the same variable (e.g., z) from both. Solve the resulting 2x2 system for x and y, then substitute back to find z.
Can simultaneous equations have no solution?
Yes. If the two equations represent parallel lines (they have the same gradient but different y-intercepts), they will never meet, meaning there is no solution. Mathematically, you end up with something impossible like 0 = 5.
What does "infinite solutions" mean?
Infinite solutions occur when the two equations represent the same line. This happens if one equation is just a multiple of the other (e.g., x + y = 2 and 2x + 2y = 4). Every point on the line is a solution.
What is Cramer's Rule?
Cramer's Rule is a matrix-based method for solving linear systems. It uses determinants of the coefficient matrix and modified matrices to find each variable directly. It is efficient for 2x2 and 3x3 systems.
How do I check if my answers are correct?
Always substitute your values for x and y back into BOTH original equations. If the calculations result in the correct constants on the right side for both equations, your answers are correct.
Are simultaneous equations used in real life?
Yes! They are used in business for break-even analysis, in physics for circuit analysis and force balances, and in chemistry for balancing chemical equations and mixture problems.
Is this solver free for GCSE revision?
Absolutely! Our solver is free and shows full step-by-step working, making it perfect for GCSE and A-Level students to check their homework and understand the solving process.
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