https://file.aiquickdraw.com/mathbot/file/1729806071942j4zc2zy6.jpg

Let's analyze the system of linear equations:

\[
\begin{cases} 
y = -\frac{1}{2}x - 3 \\
2x + y = 0
\end{cases}
\]

### Step 1: Graphing the First Equation \( y = -\frac{1}{2}x - 3 \)

This is a linear equation in slope-intercept form, \( y = mx + b \), where:
- The slope \( m = -\frac{1}{2} \)
- The y-intercept \( b = -3 \), meaning the line crosses the y-axis at \( (0, -3) \).

Using the slope \( m = -\frac{1}{2} \), we move down 1 unit for every 2 units we move to the right. This gives us the following points to graph:
- Starting at \( (0, -3) \)
- Another point can be \( (2, -4) \), obtained by moving 2 units right and 1 unit down.

### Step 2: Graphing the Second Equation \( 2x + y = 0 \)

We can rewrite this equation in slope-intercept form:
\[
y = -2x
\]
Here:
- The slope \( m = -2 \)
- The y-intercept \( b = 0 \), meaning the line passes through the origin \( (0, 0) \).

Using the slope \( m = -2 \), we move down 2 units for every 1 unit we move to the right. This gives the following points:
- Starting at \( (0, 0) \)
- Another point can be \( (1, -2) \).

### Step 3: Finding the Intersection Point

To solve the system algebraically, let's substitute \( y = -2x \) into the first equation:

\[
-2x = -\frac{1}{2}x - 3
\]

Multiply the entire equation by 2 to eliminate the fraction:

\[
-4x = -x - 6
\]

Simplify:

\[
-4x + x = -6 \quad \Rightarrow \quad -3x = -6
\]

Solve for \( x \):

\[
x = 2
\]

Substitute \( x = 2 \) into one of the original equations, say \( y = -2x \):

\[
y = -2(2) = -4
\]

Thus, the solution to the system is \( (2, -4) \).

### Step 4: Conclusion

The system has **one solution**, and the lines intersect at the point \( (2, -4) \).

### Points for Graphing:
- For \( y = -\frac{1}{2}x - 3 \): Points are \( (0, -3) \) and \( (2, -4) \).
- For \( 2x + y = 0 \): Points are \( (0, 0) \) and \( (1, -2) \).

Would you like a detailed graph of the lines or any further explanation?

#### Relative questions:
1. How do you graph a system of equations if both equations have different slopes?
2. What does it mean when two lines have the same slope but different intercepts?
3. How would you verify the solution algebraically using substitution or elimination?
4. Can you explain how the slope influences the direction of the line?
5. What happens if the lines in a system are parallel?

#### Tip: 
To check if a system of equations has a unique solution, compare the slopes. If the slopes are different, the system will have one solution.

Graph the system below and write its solution:

\[
\begin{cases} 
y = -\frac{1}{2}x - 3 \\
2x + y = 0
\end{cases}
\]

Note that you can also answer 'No solution' or 'Infinitely many solutions'.

Learn how to solve a system of linear equations by graphing. This example covers the equations y = -1/2x - 3 and 2x + y = 0, demonstrating step-by-step graphing techniques to find the solution. Ideal for Grades 8-10 students learning algebra concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation