https://file.aiquickdraw.com/mathbot/file/1729994327898q2g0miph.jpg

To solve this system of equations, we are given:

\[
y = 2x - 5
\]
\[
x^2 + y^2 = 10
\]

Let's solve this step-by-step.

1. **Substitute \( y = 2x - 5 \) into \( x^2 + y^2 = 10 \):**
   \[
   x^2 + (2x - 5)^2 = 10
   \]

2. **Expand \((2x - 5)^2\):**
   \[
   x^2 + (4x^2 - 20x + 25) = 10
   \]
   \[
   5x^2 - 20x + 25 = 10
   \]

3. **Simplify the equation:**
   \[
   5x^2 - 20x + 15 = 0
   \]
   \[
   x^2 - 4x + 3 = 0
   \]

4. **Factor the quadratic equation \( x^2 - 4x + 3 = 0 \):**
   \[
   (x - 1)(x - 3) = 0
   \]
   So, \( x = 1 \) or \( x = 3 \).

5. **Find corresponding \( y \)-values for each \( x \):**
   - For \( x = 1 \):
     \[
     y = 2(1) - 5 = -3
     \]
     So, one solution is \( (1, -3) \).
   
   - For \( x = 3 \):
     \[
     y = 2(3) - 5 = 1
     \]
     So, another solution is \( (3, 1) \).

6. **Final Answer:**
   The solutions are \( (1, -3) \) and \( (3, 1) \).

The correct answer is:
**A. \( (1, -3) \) and \( (3, 1) \)**.

Would you like more details, or have any questions?

Here are some related questions:
1. How do you solve systems of equations using substitution?
2. What are other methods to solve systems of equations besides substitution?
3. How do you determine if a solution to a system of equations is unique?
4. What does it mean geometrically when solving a system of nonlinear equations?
5. How can you use graphing to find solutions to nonlinear systems?

**Tip:** Always verify your solutions by substituting them back into the original equations to ensure accuracy.

Determine the solutions to the system of equations:

y = 2x - 5

x^2 + y^2 = 10

Step-by-step solution for determining the solutions to a system of equations using substitution. This math problem, suitable for Grades 10-11, covers key concepts in systems of equations, substitution, and quadratic 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