Question

Solve the following system of equations graphically on the set of axes below.
y, equals, start fraction, 5, divided by, 2, end fraction, x, minus, 6
y=
2
5
​
x−6
x, plus, y, equals, 8
x+y=8
Plot two lines by clicking the graph.
Click a line to delete it.

Answer 1

To solve the system of equations graphically, we will first rewrite the equations and plot them.

1. The first equation is given as:
\[
y = \frac{5}{2}x - 6
\]
This is in slope-intercept form \(y = mx + b\), where the slope \(m = \frac{5}{2}\) and the y-intercept \(b = -6\).

2. The second equation is:
\[
x + y = 8
\]
We can rewrite it in slope-intercept form:
\[
y = -x + 8
\]
Here, the slope \(m = -1\) and the y-intercept \(b = 8\).

### Steps to Graph:

#### First Equation: \(y = \frac{5}{2}x - 6\)
- Find the y-intercept: \( (0, -6) \)
- Find another point by choosing \(x = 2\):
\[
y = \frac{5}{2}(2) - 6 = 5 - 6 = -1 \quad \text{(Point: (2, -1))}
\]

#### Second Equation: \(y = -x + 8\)
- Find the y-intercept: \( (0, 8) \)
- Find another point by choosing \(x = 4\):
\[
y = -4 + 8 = 4 \quad \text{(Point: (4, 4))}
\]

### Plotting the Lines:
You can plot the first line using the points \((0, -6)\) and \((2, -1)\), and the second line using the points \((0, 8)\) and \((4, 4)\).

### Finding the Intersection Point:
- To find the intersection graphically, look for the point where both lines intersect on the graph.

Algebraically, to find the intersection:
1. Set the equations equal:
\[
\frac{5}{2}x - 6 = -x + 8
\]
2. Solve for \(x\):
\[
\frac{5}{2}x + x = 8 + 6
\]
\[
\frac{7}{2}x = 14
\]
\[
x = 4
\]
3. Now substitute \(x = 4\) back into either equation to find \(y\):
\[
y = -4 + 8 = 4
\]

Thus, the solution to the system of equations is \( (4, 4) \).

You can now plot the points and lines on a graph to visualize this solution. If you have graphing software or a graphing calculator, you can easily plot these equations to confirm the intersection point graphically.