Math Problem Statement

solve

Solve these simultaneous equations and a word problem about a trip to a theme park.

Solution

Let's start by solving the given simultaneous equations.

Problem 7: Solve these simultaneous equations by subtracting them.

(a)

x + 4y &= 11 \\ x + y &= 5 \end{align*}$$ **Step 1:** Subtract the second equation from the first: $$(x + 4y) - (x + y) = 11 - 5$$ $$x + 4y - x - y = 6$$ $$3y = 6 \quad \Rightarrow \quad y = 2$$ **Step 2:** Substitute $$y = 2$$ into the second equation: $$x + 2 = 5 \quad \Rightarrow \quad x = 3$$ **Solution:** $$x = 3$$, $$y = 2$$ --- #### (b) $$\begin{align*} 2x + 5y &= 13 \\ 2x + y &= 9 \end{align*}$$ **Step 1:** Subtract the second equation from the first: $$(2x + 5y) - (2x + y) = 13 - 9$$ $$2x + 5y - 2x - y = 4$$ $$4y = 4 \quad \Rightarrow \quad y = 1$$ **Step 2:** Substitute $$y = 1$$ into the second equation: $$2x + 1 = 9 \quad \Rightarrow \quad 2x = 8 \quad \Rightarrow \quad x = 4$$ **Solution:** $$x = 4$$, $$y = 1$$ --- #### (c) $$\begin{align*} 4x - 2y &= 16 \\ 2x - 2y &= 9 \end{align*}$$ **Step 1:** Subtract the second equation from the first: $$(4x - 2y) - (2x - 2y) = 16 - 9$$ $$4x - 2y - 2x + 2y = 7$$ $$2x = 7 \quad \Rightarrow \quad x = \frac{7}{2}$$ **Step 2:** Substitute $$x = \frac{7}{2}$$ into the second equation: $$2\left(\frac{7}{2}\right) - 2y = 9 \quad \Rightarrow \quad 7 - 2y = 9$$ $$-2y = 9 - 7 \quad \Rightarrow \quad -2y = 2 \quad \Rightarrow \quad y = -1$$ **Solution:** $$x = \frac{7}{2}$$, $$y = -1$$ --- Would you like me to proceed with problems 8, 9, or 10, or explain any specific step further? Here are five relative questions that could expand your understanding: 1. What is the elimination method in solving simultaneous equations? 2. How does subtracting two equations help in solving them? 3. Can simultaneous equations always be solved by elimination or substitution? 4. How do you solve simultaneous equations when dealing with fractions? 5. What happens if the two equations in a system are identical or parallel? **Tip:** When subtracting equations, ensure all terms are lined up (i.e., variables with variables, constants with constants) for easier simplification.

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Math Problem Analysis

Mathematical Concepts

Simultaneous Equations
Addition and Subtraction of Equations
Word Problems

Formulas

Linear equations: ax + by = c
Methods for solving simultaneous equations (addition, subtraction, multiplication)

Theorems

Elimination method for solving simultaneous equations

Suitable Grade Level

Grades 8-10

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