Linear and quadratic simultaneous equations
| English | 中文 | Pinyin |
|---|---|---|
| simultaneous solution/ˌsɪməlˈteɪnɪəs səˈluːʃn/ | 联立解 | lián lì jiě |
How many places can a line meet a parabola?
- A line can cut a parabola twice, touch it once or miss it. An algebraic solution should give the same number of real intersection points as the sketch.
- This lesson studies simultaneous solution 联立解: An ordered pair that satisfies every equation in a system at the same time.
Choose the mathematical structure
- For two linear equations, eliminate one unknown or substitute an expression for it. For one linear and one quadratic equation, express y from the line and substitute into the quadratic. Solve every resulting root, find its matching y and check 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 simultaneous solution?
An ordered pair that satisfies every equation in a system at the same time.
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 3x+2y=13 and x−y=1, add twice the second equation to the first: 5x=15, so x=3 and y=2. For x+y=4 and y=x²−2, substitute y=4−x: x²+x−6=0=(x−2)(x+3). The solutions are (2,2) and (−3,7). The line y=4x−6 instead gives (x−2)²=0, so only (2,2): a tangent. The line y=4x−7 gives x²−4x+5=0 with discriminant −4, so no real intersection.
Linear and quadratic simultaneous equations
For two linear equations, eliminate one unknown or substitute an expression for it
Connect the algebraic roots to their paired coordinates and graphical intersection counts.
For x+y=4 and y=x²−2, find the positive x solution.
Factor x²+x−6=(x−2)(x+3); the positive root is 2.
Test a tempting shortcut
- A root for x is not a complete simultaneous solution. Pair each x with its own y; do not combine one root with another root’s output. A repeated quadratic root gives one point, not two different points.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every system of two equations has exactly one real simultaneous solution. This claim is false. Explain which definition or assumption it violates.
For the same system, find y when x=−3.
Substitute x=−3: y=9−2=7, and −3+7=4.
Every system of two equations has exactly one real simultaneous solution.
A root for x is not a complete simultaneous solution. Pair each x with its own y; do not combine one root with another root’s output. A repeated quadratic root gives one point, not two different points.
Interpret a new situation
- Use the discriminant to distinguish two, one or no real intersections. The sketch helps predict the result, but substitutions into both original equations verify it. Preserve any restrictions if a more general system involves fractions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many distinct real intersection points do y=x²−2 and y=4x−6 have?
The reduced equation is (x−2)²=0; the repeated root gives one distinct point.
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
- 7357 · A-level · B. 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.
An ordered pair that satisfies every equation in a system at the same time. Choose the relationship, show the method, check its assumptions and interpret the result.