https://r2.erweima.ai/mathbot/image/1723957982277zzmlbtyk.jpg

The image you uploaded contains a system of equations that are graphed. Let me extract the mathematical content from the image and help you analyze the system of equations.

### System of Equations:
The image shows the following system of equations:
1. \( 4x - 2y = 8 \)
2. \( 2y - y = 4 \)

### Graphing the Equations:
To solve the system of equations by graphing, we need to rewrite each equation in slope-intercept form (\( y = mx + b \)), and then analyze the graph.

#### Equation 1: \( 4x - 2y = 8 \)
- Rearrange to solve for \( y \):
  \[
  -2y = -4x + 8
  \]
  \[
  y = 2x - 4
  \]

#### Equation 2: \( 2x - y = 4 \)
- Rearrange to solve for \( y \):
  \[
  -y = -2x + 4
  \]
  \[
  y = 2x - 4
  \]

### Graph Analysis:
- Both equations simplify to the same line: \( y = 2x - 4 \).
- Since the equations represent the same line, the system has infinitely many solutions. Every point on the line \( y = 2x - 4 \) is a solution.

The graph in your image should depict a single line, representing both equations, confirming that the system has infinitely many solutions.

### Questions for Further Exploration:
1. How do you interpret the solution of a system of equations when the two equations represent the same line?
2. What is the significance of the slope and y-intercept in the equation of a line?
3. How would the graph differ if the lines were parallel but distinct?
4. Can you determine the solution of a system of equations algebraically, and how does it compare to the graphical method?
5. How would the system of equations look if there were exactly one solution (the lines intersect at a single point)?

**Tip:** When graphing a system of equations, always consider the slope and intercept to quickly identify if the lines are parallel, identical, or intersecting. This will help you determine the type of solution the system has.

Explore how to analyze a system of equations graphically, focusing on understanding the solutions when equations represent the same line. Detailed step-by-step analysis and graphical interpretation included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation