Solving Equations and Inequalities. 31. The system of equations {3x – 5y = 2 and –6x + 10y = 7} has solutions.
Select . . . ▾
A. 0
B. 1
C. infinite

Let's analyze the given system of equations to determine the number of solutions:

\[
\begin{aligned}
1.\ & 3x - 5y = 2 \quad \text{(Equation 1)} \\
2.\ & -6x + 10y = 7 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 1: Simplify the Equations
Notice that the second equation can be rewritten by dividing the entire equation by \(-2\):

\[
-6x + 10y = 7 \quad \Rightarrow \quad 3x - 5y = -\frac{7}{2} \quad \text{(Simplified Equation 2)}
\]

### Step 2: Compare the Equations
Now compare the simplified version of Equation 2 with Equation 1:

- **Equation 1:** \(3x - 5y = 2\)
- **Simplified Equation 2:** \(3x - 5y = -\frac{7}{2}\)

These two equations have the same coefficients for \(x\) and \(y\) but different constants on the right-hand side (2 and \(-\frac{7}{2}\)). This indicates that the lines represented by these equations are parallel but distinct.

### Conclusion
Parallel lines never intersect, meaning there are no solutions to this system of equations.

### Answer: **A. 0**

Would you like to know more about why parallel lines result in no solutions, or have any other questions? Here are some related questions you might consider:

1. How can you algebraically determine if two lines are parallel?
2. What is the significance of the coefficients in determining the relationship between two lines?
3. Can a system of linear equations ever have exactly one solution if the equations represent parallel lines?
4. What are some real-world scenarios where solving a system of equations would be necessary?
5. How can graphing help visualize the number of solutions to a system of equations?

**Tip:** When analyzing systems of linear equations, always check if the equations are multiples of each other to determine the relationship between the lines they represent.

Explore why the system of equations 3x – 5y = 2 and –6x + 10y = 7 has no solutions. Learn about the concept of parallel lines in systems of linear equations. Suitable for high school students studying systems of equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation