Question

Solve the following system of equations graphically on the set of axes below. y, equals, minus, one half, x, plus, 8 y=− 2 1 ​ x+8 y, equals, 2, x, minus, 7 y=2x−7

Answer 1

To solve the system of equations graphically, we'll plot the two equations on the same set of axes.

1. **The first equation** is:
\[
y = -\frac{1}{2}x + 8
\]
- This is a linear equation in slope-intercept form (\(y = mx + b\)), where the slope \(m = -\frac{1}{2}\) and the y-intercept \(b = 8\).
- We can find another point to graph it. For instance, let \(x = 0\):
\[
y = -\frac{1}{2}(0) + 8 = 8 \quad \text{(Point: (0, 8))}
\]
- Let \(x = 8\):
\[
y = -\frac{1}{2}(8) + 8 = 4 \quad \text{(Point: (8, 4))}
\]
- This gives us two points: (0, 8) and (8, 4).

2. **The second equation** is:
\[
y = 2x - 7
\]
- The slope \(m = 2\) and y-intercept \(b = -7\).
- Let's find a couple of points for this line as well. For \(x = 0\):
\[
y = 2(0) - 7 = -7 \quad \text{(Point: (0, -7))}
\]
- For \(x = 5\):
\[
y = 2(5) - 7 = 3 \quad \text{(Point: (5, 3))}
\]
- This gives us two points: (0, -7) and (5, 3).

3. **Plotting the points**:
- Plot the points (0, 8) and (8, 4) for the first line and then draw a straight line through these points.
- Plot the points (0, -7) and (5, 3) for the second line and draw a straight line through these points.

4. **Finding the intersection**:
- The solution to the system of equations is at the intersection of these two lines.

To find this point algebraically, set the equations equal to each other:
\[
-\frac{1}{2}x + 8 = 2x - 7
\]

Now, solve for \(x\):
\[
8 + 7 = 2x + \frac{1}{2}x
\]
\[
15 = 2.5x
\]
\[
x = \frac{15}{2.5} = 6
\]

Now substitute \(x = 6\) back into one of the original equations to find \(y\):
\[
y = 2(6) - 7 = 12 - 7 = 5
\]

Therefore, the solution to the system of equations is:
\[
(x, y) = (6, 5)
\]

The intersection point (6, 5) can be found graphically where the two lines meet on the graph.

Answer 2

try again

Answer 3

Let's go through the process step-by-step to graphically solve the system of equations correctly.

### Given Equations
1. **First Equation:**
\[
y = -\frac{1}{2}x + 8
\]

2. **Second Equation:**
\[
y = 2x - 7
\]

### Step 1: Convert Equations into Points

#### For the first equation \(y = -\frac{1}{2}x + 8\):

- **Finding points:**
- **When \(x = 0\)**:
\[
y = -\frac{1}{2}(0) + 8 = 8 \quad \text{(Point: (0, 8))}
\]

- **When \(x = 8\)**:
\[
y = -\frac{1}{2}(8) + 8 = -4 + 8 = 4 \quad \text{(Point: (8, 4))}
\]

#### For the second equation \(y = 2x - 7\):

- **Finding points:**
- **When \(x = 0\)**:
\[
y = 2(0) - 7 = -7 \quad \text{(Point: (0, -7))}
\]

- **When \(x = 5\)**:
\[
y = 2(5) - 7 = 10 - 7 = 3 \quad \text{(Point: (5, 3))}
\]

### Step 2: Graph the Points

- **Graph the first equation** using the points: (0, 8) and (8, 4).
- **Graph the second equation** using the points: (0, -7) and (5, 3).

### Step 3: Determine the Intersection Point

To find the intersection point algebraically:

Set the two equations equal to one another:
\[
-\frac{1}{2}x + 8 = 2x - 7
\]

### Step 4: Solve for \(x\)

1. **Add \( \frac{1}{2}x \) and subtract 8 from both sides:**
\[
8 + 7 = 2x + \frac{1}{2}x
\]
\[
15 = 2.5x
\]

2. **Solve for \(x\)**:
\[
x = \frac{15}{2.5} = 6
\]

### Step 5: Find \(y\)

Now substitute \(x = 6\) back into either of the original equations; we'll use the second equation:
\[
y = 2(6) - 7 = 12 - 7 = 5
\]

### Conclusion

The intersection point, where the two lines meet, is:
\[
(x, y) = (6, 5)
\]

This point (6, 5) is the solution to the system of equations. You can plot the points and lines on a graph to visualize where they intersect. If you have graphing software or tools, you can plot both lines to verify that they intersect at (6, 5).