Two linear simultaneous equations · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| elimination/ɪˌlɪmɪˈneɪʃn/ | 消元法 | xiāo yuán fǎ |
Can one equation determine two counts?
- Two ticket types raise a total amount. A single equation cannot identify both unknown counts.
- This lesson studies elimination 消元法: Combining equations to remove one variable while retaining the same solutions.
Choose the mathematical structure
- Multiply equations to make a variable cancel, or substitute an expression from one equation into the other. Solve the remaining linear equation and recover the second variable. The intersection is a point satisfying both original equations.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines elimination?
Combining equations to remove one variable while retaining the same solutions.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x+y=12 and 3x+2y=31, subtract twice the first equation from the second: x=7. Then y=5. Check both 7+5=12 and 3×7+2×5=31. The two straight-line graphs meet at (7,5).
Two linear simultaneous equations
Multiply equations to make a variable cancel, or substitute an expression from one equation into the other
Compare the model with the worked case and explain one change.
Solve x+y=12 and 3x+2y=31. Find x.
Subtract twice x+y=12 from 3x+2y=31: x=31-24=7.
Test a tempting shortcut
- Check the ordered pair in both original equations. Multiplying an equation means multiplying every term, including the constant. Parallel distinct lines have no common solution.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A pair of simultaneous equations is solved by checking just one of them. This claim is false. Explain which definition or assumption it violates.
For those equations, find y.
Use x+y=12, so y=12-7=5.
A pair of simultaneous equations is solved by checking just one of them.
Check the ordered pair in both original equations. Multiplying an equation means multiplying every term, including the constant. Parallel distinct lines have no common solution.
Interpret a new situation
- AQA A19 Foundation solves two linear equations by elimination or substitution and interprets the graph intersection. Linear/quadratic systems belong to Higher.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
In x+y=12, find y when x=7.
The first equation gives x+y=12; substitute x=7 to obtain y=5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Combining equations to remove one variable while retaining the same solutions. Choose the relationship, show the method, check its assumptions and interpret the result.