Math Problem Statement

How many solutions (x, y) does the given system of equations have?

Solution

The system of equations provided is:

1) \quad 5x + 5y = 15 \\ 2) \quad 14y - 28x = 42 \end{aligned}$$ We need to determine how many solutions $$(x, y)$$ this system has. Here's the step-by-step process: ### Step 1: Simplify both equations - First equation: $$5x + 5y = 15$$ can be simplified by dividing through by 5: $$x + y = 3 \quad \text{(Equation 1)}$$ - Second equation: $$14y - 28x = 42$$ can be simplified by dividing through by 14: $$y - 2x = 3 \quad \text{(Equation 2)}$$ ### Step 2: Solve the system Now we solve this simplified system of equations: $$\begin{aligned} 1) \quad x + y = 3 \\ 2) \quad y - 2x = 3 \end{aligned}$$ - From the first equation: $$y = 3 - x$$. - Substitute this into the second equation: $$(3 - x) - 2x = 3$$ Simplify: $$3 - x - 2x = 3 \quad \Rightarrow \quad 3 - 3x = 3$$ Subtract 3 from both sides: $$-3x = 0 \quad \Rightarrow \quad x = 0$$ - Substitute $$x = 0$$ into the first equation $$x + y = 3$$: $$0 + y = 3 \quad \Rightarrow \quad y = 3$$ Thus, the system has exactly **one solution**, which is $$(x, y) = (0, 3)$$. ### Conclusion: The system of equations has **exactly one solution**, so the correct answer is **B: Exactly One**. Would you like any further details or explanations? Here are 5 related questions to expand on this topic: 1. What are the different types of solutions a system of linear equations can have? 2. How do you solve a system of equations using substitution versus elimination? 3. What is the geometric interpretation of a system with exactly one solution? 4. How do we recognize when a system of equations has no solution? 5. Can you explain how to verify the solution of a system of equations? **Tip:** Always simplify equations as much as possible before solving—it makes the problem easier to handle!

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Solution of Equations

Formulas

Linear equation in two variables: ax + by = c
Solving a system by substitution or elimination

Theorems

Consistent and Inconsistent Systems
Unique Solution Theorem

Suitable Grade Level

Grades 8-10