Linear and linear/quadratic simultaneous equations · Higher
導入済み| English | 意味 |
|---|---|
| elimination/ɪˌlɪmɪˈneɪʃn/ | 消元法 |
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
- For two linear equations, use elimination or substitution and check both equations. For a line and a quadratic, substitute the linear relation first; then solve the resulting quadratic. For inequalities, shade the region satisfying every condition.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
After finding x=7 from x+y=12 and 3x+2y=31, why must the answer be checked in BOTH equations?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Elimination creates derived equations; an arithmetic slip can satisfy a derived equation while failing an original one. Substituting (7,5): 7+5=12 and 21+10=31 - both hold.
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 to obtain x=7,y=5. For y=x+2 and y=x², equate outputs: x²-x-2=0, hence x=2 or -1. The intersections are (2,4) and (-1,1), and both satisfy the line and parabola.
Linear and linear/quadratic simultaneous equations
For two linear equations, use elimination or substitution and check both equations
Compare the model with the worked case and explain one change.
Solve x+y=12 and 3x+2y=31. Find x.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Subtract twice x+y=12 from 3x+2y=31: x=31-24=7.
Test a tempting shortcut
- One equation checked is not enough. A line can meet a quadratic twice, so retain both solutions unless the context removes one. Inequality boundaries may be included or excluded according to the sign.
- 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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Use x+y=12, so y=12-7=5.
A pair of simultaneous equations is solved by checking just one of them.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
One equation checked is not enough. A line can meet a quadratic twice, so retain both solutions unless the context removes one. Inequality boundaries may be included or excluded according to the sign.
Interpret a new situation
- AQA A19 Higher includes two linear equations and linear/quadratic systems. Elimination suits linear pairs; substitution reduces a line/curve pair to a quadratic. Retain every solution and check both original equations.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the larger x-coordinate where y=x+1 meets y=x²-1.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Equate the graphs: x²-x-2=(x-2)(x+1)=0. The larger root is 2.
Match each solving method to its situation.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Line-and-quadratic pairs can meet twice, so keep both algebraic solutions unless the context excludes one.
Use this in your course
- 8300 · Higher · 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.