Simultaneous Showdown

Simultaneous Showdown 🤠

**Simultaneous equations** are two equations that share the **same** two unknowns. The solution is the one pair of values (x, y) that makes **both** true at once. The classic method is **elimination**: get rid of one variable so you can solve for the other. Time for the showdown.

The elimination idea ⚖️

If a variable has the **same coefficient** in both equations, you can remove it by adding or subtracting the equations. Take **x + y = 5** and **x − y = 1**. The y terms are +y and −y — opposites — so **adding** the equations cancels y: (x + y) + (x − y) = 5 + 1, giving 2x = 6.

Worked example 📋

Solving **x + y = 5** and **x − y = 1**: 1. **Add** the equations: 2x = 6, so **x = 3**. 2. **Back-substitute** into x + y = 5: 3 + y = 5, so **y = 2**. 3. **Check** in the other equation: 3 − 2 = 1. ✓\n\nThe solution is x = 3, y = 2.

Order the method

An interactive activity.

Solve for x

An interactive activity.

Solve for y

An interactive activity.

Add or subtract? ➗

Which operation eliminates the variable depends on the **signs** of its matching coefficients: • **Same sign** (both +y, or both −y) → **subtract** the equations. • **Opposite signs** (+y and −y) → **add** the equations. A quick memory aid: **S**ame **S**ubtract. If the coefficients do not match yet, multiply an equation through first to make them match.

Which operation?

For 2x + y = 7 and x − y = 2, you want to eliminate y. The y terms are +y and −y. Do you add or subtract the equations?

• Add them — the signs are opposite, so adding cancels y
• Subtract them
• Multiply the two equations together
• Neither — solve each on its own

Check the solution

The solution to x + y = 5 and x − y = 1 is x = 3, y = 2. Pick the TWO statements that show it is correct.

• x + y = 5 works: 3 + 2 = 5
• x − y = 1 works: 3 − 2 = 1
• x + y = 6 works: 3 + 2 = 6
• The solution is really x = 2, y = 3

The graphical showdown 📉

Each linear equation is a **straight line**. The solution to a pair of simultaneous equations is exactly the **point where the two lines cross** — the one (x, y) that lies on both. For x + y = 5 and x − y = 1, the lines cross at (3, 2) — the same answer elimination gave.

Plot the solution

An interactive activity.

The showdown rules

To solve linear simultaneous equations by elimination, make one variable's _____ match, then add or subtract to remove it. On a graph, the solution is the point where the two lines _____. Always _____ your answer in both equations.