Simultaneous Linear Equations

1Summary 重點摘要

Solve two linear equations, use substitution and elimination.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Write both equations clearlyTo solve a pair of simultaneous linear equations, first write both equations neatly, ordered in x and y, and label them (1) and (2) on the right so they can be referred to later. Labelling looks trivial but prevents quoting the wrong equation while substituting or eliminating. If an equation contains brackets or fractions, expand and simplify it first. A common mistake is miscopying a coefficient or a sign, which ruins every later calculation.
2. Step 2: Rearrange for substitutionThe first move of the substitution method is to pick one equation and express one variable in terms of the other, for example x + y = 5 gives x = 5 - y. Choose the variable with the simplest coefficient, ideally 1, so the rearrangement is least error-prone. The rule behind rearranging is that subtracting equal amounts from both sides keeps an equation true. A common mistake is forgetting the sign flip, for example writing x + y = 5 as x = 5 + y.
3. Step 3: Substitute to get one equationSubstitute the rearranged expression into the other equation: putting x = 5 - y into x - y = 1 gives (5 - y) - y = 1, leaving a single linear equation in y. The key idea is elimination: substitution turns two unknowns into one. Simplify term by term and collect like terms to get y = 2. A common mistake is a sign slip when collecting the y terms, for example simplifying (5 - y) - y as 5 - y + y.
4. Step 4: Substitute back for the other variableOnce y is known, substitute its value into the rearranged expression from earlier, for example x = 5 - y gives x = 5 - 2 = 3. Substitute into the rearranged form rather than an original equation, because x already stands alone there, which is the most direct and least error-prone route. The pair of x and y values is the solution of the system. A common mistake is putting the value into the wrong equation or copying a sign wrongly.
5. Step 5: Elimination methodWhen the coefficients are tidy, elimination by adding or subtracting is often faster: first make the coefficients of one variable equal or opposite by multiplying each equation by a suitable number, then add or subtract the equations to remove that variable. For example, adding x + y = 5 and x - y = 1 cancels the y terms, giving 2x = 6, so x = 3. A common mistake is multiplying only the left side of an equation, or mis-handling signs when one coefficient is negative.
6. Step 6: Check the solutionFinally substitute the found x and y into both original equations; the pair is correct only when both hold, since checking just one equation can miss a substitution error that happens to satisfy it. For the example values x = 3 and y = 2, both x + y = 5 and x - y = 1 hold, so the answer is reliable. Checking costs a few seconds in an exam yet blocks most arithmetic slips. A common mistake is checking a simplified form instead of the original equations.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Simultaneous equations have two variables
• Substitution: one equation express one variable, substitute into another
• Elimination: multiply suitable, add/subtract eliminate one variable
• Both variables must be found
• Answer must substitute to check