Simultaneous equations and feasible regions · Higher
| 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
- 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.
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.
If x+y=12 and 3x+2y=31, subtract twice the first equation from the second to obtain x=7, then y=5. For y=x+1 and y=x²-1, x²-x-2=0 gives x=2 or -1, with y=3 or 0.
Simultaneous equations and feasible regions
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.
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.
Use x+y=12, so y=12-7=5.
A pair of simultaneous equations is solved by checking just one of them.
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
- Define the variables and their units. If they count objects, both must be nonnegative integers. For a feasible region, test one point on the required side of each boundary and then take the intersection.
- 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.
Equate the graphs: x²-x-2=(x-2)(x+1)=0. The larger root is 2.
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
- 4MA1 · Higher · 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.