Math Problem Statement

Determine the solution to each system of equations and identify three points that are solutions to each inequality system.

Solution

Let's solve the given problems step by step.

Problem 2(a): Solve the system of equations

-3x - 4y = -13$$ **Step 1: Simplify the system.** From the first equation: $$6x + 2y = 2 \quad \Rightarrow \quad 3x + y = 1 \quad \text{(divide by 2)}.$$ The system becomes: $$3x + y = 1 \quad \text{(1)} \\ -3x - 4y = -13 \quad \text{(2)}.$$ **Step 2: Solve by elimination.** Add equations (1) and (2): $$(3x + y) + (-3x - 4y) = 1 - 13 \\ -3y = -12 \quad \Rightarrow \quad y = 4.$$ Substitute $$y = 4$$ into equation (1): $$3x + 4 = 1 \quad \Rightarrow \quad 3x = -3 \quad \Rightarrow \quad x = -1.$$ **Solution**: $$(x, y) = (-1, 4)$$. --- ### Problem 2(b): Solve the system of equations $$0.5x - 0.25y = -0.5 \\ 0.2x + 0.75y = 6.6$$ **Step 1: Eliminate decimals.** Multiply the first equation by 4 and the second by 10: $$2x - y = -2 \quad \text{(3)} \\ 2x + 7.5y = 66 \quad \text{(4)}.$$ **Step 2: Solve by elimination.** Subtract equation (3) from equation (4): $$(2x + 7.5y) - (2x - y) = 66 - (-2) \\ 8.5y = 68 \quad \Rightarrow \quad y = 8.$$ Substitute $$y = 8$$ into equation (3): $$2x - 8 = -2 \quad \Rightarrow \quad 2x = 6 \quad \Rightarrow \quad x = 3.$$ **Solution**: $$(x, y) = (3, 8)$$. --- ### Problem 3(a): Find three points satisfying the inequalities $$y \geq 0.75x \\ y \leq 2x - 7$$ **Step 1: Graphically analyze the region.** The inequalities represent a region where: - $$y \geq 0.75x$$: Above the line $$y = 0.75x$$. - $$y \leq 2x - 7$$: Below the line $$y = 2x - 7$$. **Step 2: Test points within the region.** Some solutions are: 1. $$(4, -1)$$, 2. $$(6, 3)$$, 3. $$(8, 5)$$. --- ### Problem 3(b): Find three points satisfying the inequalities $$y < 0.5x - 2 \\ y > -0.25x + 3$$ **Step 1: Graphically analyze the region.** The inequalities represent a region where: - $$y < 0.5x - 2$$: Below the line $$y = 0.5x - 2$$. - $$y > -0.25x + 3$$: Above the line $$y = -0.25x + 3$$. **Step 2: Test points within the region.** Some solutions are: 1. $$(2, 0)$$, 2. $$(4, -1)$$, 3. $$(6, 1)$$. --- ### Summary of Solutions 1. **2(a)**: $$(-1, 4)$$. 2. **2(b)**: $$(3, 8)$$. 3. **3(a)**: Example points: $$(4, -1)$$, $$(6, 3)$$, $$(8, 5)$$. 4. **3(b)**: Example points: $$(2, 0)$$, $$(4, -1)$$, $$(6, 1)$$. --- Would you like me to provide more detailed steps or clarify any part of the solutions? ### Follow-Up Questions: 1. What other methods can be used to solve systems of equations (e.g., substitution)? 2. How do you graph inequalities on a coordinate plane? 3. How can we verify solutions to inequalities? 4. What is the importance of elimination vs. substitution in solving systems? 5. How do you determine boundary lines for inequalities? **Tip**: Always double-check your solutions by substituting them back into the original equations!

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Inequalities
Graphical Solution of Systems
Algebraic Manipulation

Formulas

Standard form of linear equations: Ax + By = C
Slope-intercept form of inequalities: y = mx + b

Theorems

Addition/Elimination Method for Systems of Equations
Substitution Method for Solving Systems
Graphical Representation of Linear Inequalities

Suitable Grade Level

Grades 9-12

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