Simultaneous Equations

What are Simultaneous Equations?

Simultaneous equations are two (or more) equations that are both true at the same time. The solution is the pair of values (x, y) that satisfy both equations simultaneously.

Example:

• Equation 1: x + y = 7
• Equation 2: x - y = 3
• Solution: x = 5, y = 2 (check: 5 + 2 = 7 ✓ and 5 - 2 = 3 ✓)

You cannot solve a single equation with two unknowns — you need two equations to find a unique solution.

Method 1: Elimination

Add or subtract the equations to eliminate one variable, leaving a single equation with one unknown.

Worked Example: Solve x + y = 7 and x - y = 3

Label the equations:

• x + y = 7 ... (1)
• x - y = 3 ... (2)

Add (1) + (2): 2x = 10 → x = 5

Substitute x = 5 into (1): 5 + y = 7 → y = 2

Solution: x = 5, y = 2

Check both equations: 5 + 2 = 7 ✓ and 5 - 2 = 3 ✓

When Coefficients Do Not Match

If the variable you want to eliminate has different coefficients, multiply one or both equations first.

Example: Solve 2x + y = 10 and x + y = 7

Subtract equation 2 from equation 1: (2x + y) - (x + y) = 10 - 7 → x = 3

Substitute: 3 + y = 7 → y = 4. Solution: x = 3, y = 4

Harder example: Solve 3x + 2y = 16 and x + y = 7

Multiply equation 2 by 2: 2x + 2y = 14

Subtract: (3x + 2y) - (2x + 2y) = 16 - 14 → x = 2

Substitute: 2 + y = 7 → y = 5. Solution: x = 2, y = 5

Method 2: Substitution

Rearrange one equation to express one variable in terms of the other, then substitute into the second equation.

Worked Example: Solve y = 2x and x + y = 9

Equation 1 already gives y in terms of x: y = 2x

Substitute into equation 2: x + 2x = 9 → 3x = 9 → x = 3

Substitute back: y = 2(3) = 6

Solution: x = 3, y = 6

Check: 3 + 6 = 9 ✓ and 6 = 2(3) ✓

When to Use Substitution

Substitution is easiest when one equation already has a variable isolated (like y = ... or x = ...). If both equations require rearranging, elimination is usually faster.

Method 3: Graphical Method

Plot both equations as straight lines on the same set of axes. The intersection point gives the solution.

Steps:

1. Rearrange each equation into y = mx + c form
2. Draw both lines carefully
3. Read off the coordinates of the intersection point

This method is good for visualising the solution but can be inaccurate if the intersection is not at whole-number coordinates.

Number of Solutions

Situation	Lines	Solutions
Lines cross at one point	Different gradients	One unique solution
Lines are parallel	Same gradient, different intercept	No solution
Lines are identical	Same gradient, same intercept	Infinitely many solutions

Setting Up From Word Problems

Many problems require you to form the equations first.

Example: Two numbers sum to 20. Their difference is 4. Find the numbers.

Let the numbers be x and y:

• x + y = 20 ... (1)
• x - y = 4 ... (2)

Add: 2x = 24 → x = 12, then y = 8. The numbers are 12 and 8.

Common Mistakes

• Substituting into the same equation you rearranged (always substitute into the other equation)
• Forgetting to check the solution in both equations
• Sign errors when subtracting equations — especially with negative terms
• Not multiplying both sides of an equation when scaling up coefficients

Tips and Tricks

• Always label equations (1) and (2) to stay organised
• After finding x, substitute back to find y — do not leave the answer half-finished
• Always verify your solution in both original equations
• If elimination looks messy, try substitution instead (and vice versa)
• The graphical method gives a good visual check even if you solve algebraically