פרמטרים ליניאריים, מספרי פתרונות ואי-שוויונות
הוצג| English | משמעות |
|---|---|
| identity/aɪˈdentɪti/ | זהות |
| boundary line/ˈbaʊndəri laɪn/ | קו גבול |
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?
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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.
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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 ____.
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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.
איזו תשובה מתאימה למקרה זה?
פתרון ומיון משוואות ליניאריות בمتغير אחד עם פרמטרים
The system x+y=2 and 2x+2y=5 has an intersection.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.