Linear parameters, solution counts and inequalities
| English | 中文 | Pinyin |
|---|---|---|
| identity/aɪˈdentɪti/ | 恒等式 | héng děng shì |
| boundary line/ˈbaʊndəri laɪn/ | 边界直线 | biān jiè zhí xiàn |
A decision before an answer
- An equation can simplify to 0 = 0 without telling you x = 0. The statement means every allowed value works.
- Your goal: Solve and classify one-variable linear equations with parameters.
Read the relationship
- Collect variable terms and constants separately. An equation ax+b=cx+d becomes (a-c)x=d-b. If a-c is nonzero there is one solution; if both sides reduce to zero there are infinitely many; if the variable coefficient is zero but the constant is nonzero there are none. Never divide by a parameter before checking whether it can be zero.
- Interpret two-variable linear equations as sets of ordered pairs.
For ax+4=3x+4, which a gives infinitely many solutions?
At a=3 both sides are the same expression.
Use the defining rule
- A two-variable equation gives a set of ordered pairs. In y=mx+b, m is the change in y per unit x and b is y at x=0. For a contextual model, attach units to both: a taxi charge C=3t+5 has a 5-unit initial fee and 3 units per time unit. The intercept may be outside the practical domain even when it exists algebraically.
- Interpret slope and intercept of a contextual linear function.
Solve -3x≤12.
Dividing by -3 reverses ≤ to ≥.
Check the conditions
- For a system, compare both coefficients and constants. Multiplying one whole equation by a nonzero number produces the same line; matching variable coefficients with different constants gives parallel distinct lines. Otherwise an intersection supplies the pair satisfying both equations. Substitute the pair into both original equations to check it.
- Classify two-equation linear systems and check their intersections.
For kx+6=2x+6, k=2 gives 0=0 and infinitely many solutions; k≠2 gives x=0. For -2x+3<11, subtract 3 and divide by -2 to get x>-4. For y≤2x+1, (0,0) works and the solid boundary is included. The system 2x+4y=10 and x+2y=6 has no solution because doubling the second gives the same left side equal to 12, not 10.
For y=7x+9, the y-intercept is ____.
Set x=0 to obtain y=9.
Apply the task format
- Multiplying or dividing an inequality by a negative number reverses its direction. A strict inequality excludes the boundary, while ≤ or ≥ includes it. For two variables, draw the corresponding line and test a point away from it to identify the allowed half-plane. The diagram is a representation of all solutions, not a single preferred point.
- Represent strict and inclusive inequalities in one or two variables.
Do not confuse x=0 with a vanished variable coefficient, reverse only when multiplying/dividing by a negative, or shade a half-plane from the line’s slope alone.
Which answer fits this case?
Solve and classify one-variable linear equations with parameters
The system x+y=2 and 2x+2y=5 has an intersection.
Doubling the first gives 2x+2y=4, inconsistent with 5.
Keep the distinctions
- identity 恒等式 — An equation true for every value in its stated domain.
- boundary line 边界直线 — The equality separating allowed and excluded pairs in a linear inequality.
- Solve and classify one-variable linear equations with parameters.
- Interpret two-variable linear equations as sets of ordered pairs.
- Interpret slope and intercept of a contextual linear function.
- Classify two-equation linear systems and check their intersections.
- Represent strict and inclusive inequalities in one or two variables.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.