https://file.aiquickdraw.com/mathbot/file/1726702069147tanfyxr6.jpg

The system of equations given in the image is:

\[
\begin{aligned}
2x + y &= 7 \\
y &= \frac{1}{2}x + 2
\end{aligned}
\]

We can solve this system of equations graphically and algebraically:

### Step 1: Graph both equations.

1. **First equation**: \( 2x + y = 7 \)
   - Rewrite it in slope-intercept form: \( y = -2x + 7 \).
   - The slope is \( -2 \) and the y-intercept is \( 7 \).

2. **Second equation**: \( y = \frac{1}{2}x + 2 \)
   - The slope is \( \frac{1}{2} \) and the y-intercept is \( 2 \).

### Step 2: Find the intersection.
From the graph, the lines intersect at the point \( (2, 3) \).

### Step 3: Verify the solution algebraically.
Substitute \( x = 2 \) into both equations:

1. For \( 2x + y = 7 \):
   \[
   2(2) + 3 = 7 \quad \text{(True)}
   \]

2. For \( y = \frac{1}{2}x + 2 \):
   \[
   3 = \frac{1}{2}(2) + 2 = 3 \quad \text{(True)}
   \]

Thus, the solution to the system is \( (2, 3) \).

### Final Answer:
The solution to the system is \( (2, 3) \).

Would you like more details or clarifications on any of the steps?

Here are some related questions for further exploration:
1. What is the significance of the slope in these equations?
2. How can you determine the solution to a system if the lines are parallel?
3. What happens if both equations represent the same line?
4. How do you solve systems of equations algebraically (using substitution or elimination)?
5. What are the conditions for a system of equations to have infinitely many solutions?

**Tip:** Always check your solution by substituting the values back into the original equations to ensure accuracy.

Graph the system below and write its solution. System of equations: 2x + y = 7, y = 1/2x + 2

Explore how to graphically solve a system of linear equations involving 2x + y = 7 and y = 1/2x + 2. The solution is verified algebraically and the intersection point (2,3) is identified, which is the point where both lines meet on the graph.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation