Simultaneous Linear Equations

Solution Pair

Meaning: A solution pair gives values of both variables.

Definition: A solution pair satisfies every equation in the system.

If $x = 2$ and $y = 3$ works in both equations, then (2, 3) is the solution.

• Check both equations.

Substitution Method

Meaning: One variable is replaced by an equal expression.

Definition: Solve one equation for one variable, then substitute into the other equation.

If $y = x + 1$, replace y with $x + 1$ in the second equation.

• Isolate one variable.
• Substitute.
• Solve and bac$k-s$ubstitute.

Elimination Method

Meaning: One variable is cancelled.

Definition: Add or subtract equations after making one pair of coefficients equal or opposite.

If equations contai$n +2$y an$d -2$y, adding cancels y.

• Align like terms.
• Make coefficients match if needed.
• Add or subtract carefully.

Easy

Question: Solve $x + y = 7$ and $x - y = 1$.

1. Add the equations: 2$x = 8$.
2. $x = 4$.
3. Substitute: $4 + y = 7$.
4. $y = 3$.

Medium

Question: Solve 2$x + y = 11$ and $y = 3$.

1. Substitute $y = 3$.
2. 2$x + 3 = 11$.
3. 2$x = 8$.
4. $x = 4$.

Exam-level

Question: Solve 3$x + 2$$y = 16$ and 3$x - y = 7$.

1. Subtract the second equation from the first: 3$y = 9$.
2. $y = 3$.
3. Substitute in 3$x - y = 7$.
4. 3$x - 3 = 7$.
5. $x = 10 / 3$.