1. **Main Aim:** To understand and solve simple equations and variations, including change of subject, substitution, and types of variation. 2. **Subsidiary Aim:** To develop skills in manipulating formulas and solving real-life problems involving variation. 3. **Assumption:** Students have basic algebra knowledge including operations with variables and solving linear equations. 4. **Anticipated Problems and Solutions:** - Difficulty in rearranging formulas: Use step-by-step guided examples. - Confusion between types of variation: Provide clear definitions and examples. 5. **Teaching Aids:** Whiteboard, markers, worksheets, and graphing tools. --- ### Change of Subject Formula **Problem:** Change the subject of the formula $A = 2B + 3C$ to $B$. **Step 1:** State the formula and what is required. **Step 2:** Isolate $B$ on one side: $$A = 2B + 3C$$ **Step 3:** Subtract $3C$ from both sides: $$A - 3C = 2B$$ **Step 4:** Divide both sides by 2: $$\frac{A - 3C}{2} = B$$ **Step 5:** Show cancellation: $$B = \frac{\cancel{A - 3C}}{\cancel{2}}$$ (no cancellation here, just division) **Answer:** $$B = \frac{A - 3C}{2}$$ --- ### Substitution **Problem:** Given $y = 3x + 2$ and $x = 4$, find $y$. **Step 1:** Substitute $x = 4$ into the equation: $$y = 3(4) + 2$$ **Step 2:** Calculate: $$y = 12 + 2 = 14$$ **Answer:** $$y = 14$$ --- ### Variation **Definition:** - **Direct Variation:** $y$ varies directly as $x$ means $y = kx$ where $k$ is constant. - **Inverse Variation:** $y$ varies inversely as $x$ means $y = \frac{k}{x}$. - **Joint Variation:** $y$ varies jointly as $x$ and $z$ means $y = kxz$. --- ### Solved Problems on Variation **Problem 1 (Simple):** If $y$ varies directly as $x$ and $y = 10$ when $x = 2$, find $y$ when $x = 5$. **Step 1:** Write the formula: $$y = kx$$ **Step 2:** Find $k$ using given values: $$10 = k \times 2 \Rightarrow k = \frac{10}{2} = 5$$ **Step 3:** Find $y$ when $x=5$: $$y = 5 \times 5 = 25$$ **Answer:** $$y = 25$$ --- **Problem 2 (Moderate):** $y$ varies inversely as $x$. If $y = 6$ when $x = 4$, find $y$ when $x = 12$. **Step 1:** Write the formula: $$y = \frac{k}{x}$$ **Step 2:** Find $k$: $$6 = \frac{k}{4} \Rightarrow k = 6 \times 4 = 24$$ **Step 3:** Find $y$ when $x=12$: $$y = \frac{24}{12} = 2$$ **Answer:** $$y = 2$$ --- **Problem 3 (Complex):** $y$ varies jointly as $x$ and $z$. If $y = 24$ when $x = 2$ and $z = 3$, find $y$ when $x = 4$ and $z = 5$. **Step 1:** Write the formula: $$y = kxz$$ **Step 2:** Find $k$: $$24 = k \times 2 \times 3 \Rightarrow 24 = 6k \Rightarrow k = \frac{24}{6} = 4$$ **Step 3:** Find $y$ when $x=4$ and $z=5$: $$y = 4 \times 4 \times 5 = 80$$ **Answer:** $$y = 80$$