C03 Equations
对应大纲的内容
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<td style=”width: 100px; border: none; font-weight: bold; vertical-align: top; padding: 1em 1em 1em 1em;”>M4.15</td>
<td style=”border: none; vertical-align: top; padding: 1em 1em 1em 0;”>Set up and solve, both algebraically and graphically, simple equations including simultaneous equations involving two unknowns; this may include one linear and one quadratic equation.
Solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically.
Find approximate solutions using a graph.
Translate simple situations or procedures into algebraic expressions or formulae; for example, derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.</td>
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<td style=”width: 100px; border: none; font-weight: bold; vertical-align: top; padding: 1em 1em 1em 1em;”>MM1.4</td>
<td style=”border: none; vertical-align: top; padding: 1em 1em 1em 0;”>Simultaneous equations: analytical solution by substitution, e.g. of one linear and one quadratic equation.</td>
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学习目标
- Analyse simultaneous linear systems: Determine parameter conditions for unique, infinite, or no solutions, and link them to their geometric equivalents (intersecting, coincident, or parallel lines/planes).
- Solve linear-quadratic systems: Solve mixed systems via substitution and map algebraic results to geometric intersections (secants, tangents, or disjoint curves).
- Deconstruct absolute value equations: Solve nested equations (e.g., $||f(x)|-a|=b$ ) using critical boundary points, piecewise case analysis, and graphical sketching.
- Solve irrational equations: Resolve radical terms by isolating and squaring, or by applying polynomial substitutions like $u=\sqrt{f(x)}$.
- Enforce domain restrictions: Identify and reject extraneous solutions, ensuring derived roots satisfy the original domains for square roots $(x \geq 0)$ and absolute values $(|x| \geq 0)$.
建议时间
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 16 | 140 min |
| Quiz | 7 | 15 min |
| Practice A | 30 | 60 min |
| Practice B | 20 | 75 min |
| Total | 73 | approx. 4-5 hrs |
