Simultaneous Linear Equations

1Summary 重點摘要

Two equations with two unknowns, solve for x and y by substitution or elimination.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What are simultaneous equations?A pair of simultaneous linear equations consists of two linear equations sharing the same two unknowns, usually x and y, such as x + y = 5 and x − y = 1. One equation alone has infinitely many solutions; the goal is the single pair (x, y) satisfying both at once. 'Linear' means the unknowns appear only to the first power — no x² or xy terms.
2. 2. Method 1: SubstitutionSubstitution works best when one equation is already in the form 'y = ...' or 'x = ...'. For y = x + 1 and x + y = 5: first replace y in the second equation with the whole expression (x + 1), giving x + (x + 1) = 5; then collect like terms to get 2x + 1 = 5, so 2x = 4 and x = 2. Two unknowns have temporarily become one, which can be solved directly.
3. 3. Substitution: find the other unknownFinding x = 2 is not the end: substitute it back into the simplest original equation to obtain y. From y = x + 1 we get y = 2 + 1 = 3, so the solution is x = 2, y = 3. A common mistake is stopping after x — a pair of simultaneous equations always needs both values, stated clearly, and it is worth checking them in the other equation too.
4. 4. Method 2: EliminationElimination is fastest when the two equations carry equal or opposite coefficients for one unknown. For x + y = 5 and x − y = 1, adding the equations gives (x + y) + (x − y) = 2x on the left and 5 + 1 = 6 on the right; the y terms cancel, leaving 2x = 6, so x = 3. If the coefficients differ, multiply one equation by a suitable number first to make them match, then add or subtract.
5. 5. Elimination: find the other unknownSubstitute x = 3 back into the first equation x + y = 5: 3 + y = 5 gives y = 2, so the solution is x = 3, y = 2. Choose the simplest equation for this back-substitution to keep the arithmetic light. If decimals or fractions appear without reason, an earlier elimination step is probably wrong — retrace the steps instead of forcing the wrong numbers onwards.
6. 6. Check and exam tipsCheck the pair in both original equations: x + y = 3 + 2 = 5 ✓ and x − y = 3 − 2 = 1 ✓; only when both hold is the solution confirmed, because satisfying one equation alone still leaves infinitely many pairs. For the exam: turn word problems into two equations first, and remember to answer the original question — a price or a quantity — rather than only the bare values of x and y.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Simultaneous equations = 2 equations, 2 unknowns
• Substitution: plug one into the other
• Elimination: add/subtract to cancel one unknown
• After finding x, still need to find y
• Check both equations to be safe