Linear and quadratic simultaneous equations
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| simultaneous solution/ˌsɪməlˈteɪnɪəs səˈluːʃn/ | nghiệm đồng thời |
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.
The method
- 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.
The line y=4x-6 meets y=x²-2 where (x-2)²=0. What does the repeated root mean geometrically?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A repeated root gives exactly one intersection point with the line grazing the curve. A positive discriminant would give two crossings; the negative discriminant of y=4x-7 gives none.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Factor x²+x−6=(x−2)(x+3); the positive root is 2.
Where it goes wrong
- 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.
Not true: Every system of two equations has exactly one real simultaneous solution. Two equations can have no solution, one, or several; a line and a circle can meet twice or not at all.
For the same system, find y when x=−3.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Substitute x=−3: y=9−2=7, and −3+7=4.
Every system of two equations has exactly one real simultaneous solution.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Two equations can have no solution, one, or several; a line and a circle can meet twice or not at all.
Applying it
- 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.
How many distinct real intersection points do y=x²−2 and y=4x−6 have?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The reduced equation is (x−2)²=0; the repeated root gives one distinct point.
Match each line-curve outcome to its algebraic sign.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Substitute the linear relation into the quadratic, then pair each x root with its own y - never mix roots.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Simultaneous solution: An ordered pair that satisfies every equation in a system at the same time.
- For two linear equations, eliminate one unknown or substitute an expression for it.
- Watch for: A root for x is not a complete simultaneous solution.