What You Need to Know
The Big Picture
Simultaneous equations are a pair of equations with two unknowns (usually x and y). Both equations must be satisfied by the same pair of values. There are two main types: linearquadratic (one or both contain squared terms like x² or y²). Each type has different solving methods, and you need to master all of them.
Why two equations for two unknowns?
One equation with two unknowns has infinitely many solutions (a whole line on a graph). Two equations together have a unique solution (or sometimes no solution, or infinitely many). In real life, you use simultaneous equations to solve problems like "find two numbers with specific properties" or "find the cost and quantity from two shopping scenarios".
Linear Simultaneous Equations
What Are They?
Linear simultaneous equations are two equations with two unknowns where neither equation contains squared terms (no x², y², or xy). Both equations represent straight lines. The solution is where these lines intersect on a graph. Examples: 3x + 2y = 11 and 2x − y = 5.
Method 1: Elimination
How Elimination Works
Elimination removes one variable by making its coefficient the same in both equations, then adding or subtracting. The key rule: if the signs of the term you want to eliminate are the same, subtract; if they're different, add.
Step-by-Step Process
1
Decide which variable to eliminate (x or y)
2
Multiply one or both equations so the coefficients of that variable are the same
3
Add or subtract the equations (depending on whether the signs are different or the same)
4
Solve the resulting single equation to find one variable
5
Substitute back into one of the original equations to find the other variable
6
Check both solutions in both original equations
Example 1: Elimination with Same Signs
Solve: 3x + 2y = 11 and 2x − y = 5
Multiply equation 1 by 2: 6x + 4y = 22
Multiply equation 2 by 3: 6x − 3y = 15
Now the x coefficients are both 6 (same sign).
Since both are positive, SUBTRACT:
6x + 4y = 22
−(6x − 3y = 15)
_______________
7y = 7
y = 1
Substitute y = 1 into equation 2:
2x − 1 = 5
2x = 6
x = 3
Check: 3(3) + 2(1) = 9 + 2 = 11 ✓
2(3) − 1 = 6 − 1 = 5 ✓
Method 2: Substitution
How Substitution Works
Rearrange one equation to express one variable in terms of the other (e.g., y = 2x − 5), then substitute this entire expression into the other equation. This reduces two equations to one.
1
Rearrange one equation into the form y = ... (or x = ...)
2
Substitute this expression into the other equation (replace y with the expression)
3
4