To solve the system of equations by graphing, we first need to rearrange each equation into slope-intercept form, which is [tex]\( y = mx + b \)[/tex].
### Step 1: Rewrite the Equations
Equation 1: [tex]\( x + 5y = 10 \)[/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 5y = 10 - x \][/tex]
[tex]\[ y = \frac{10 - x}{5} \][/tex]
[tex]\[ y = -\frac{1}{5}x + 2 \][/tex]
Equation 2: [tex]\( 3x + 5y = 0 \)[/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 5y = -3x \][/tex]
[tex]\[ y = -\frac{3}{5}x \][/tex]
So, we have the following equations in slope-intercept form:
1. [tex]\( y = -\frac{1}{5}x + 2 \)[/tex]
2. [tex]\( y = -\frac{3}{5}x \)[/tex]
### Step 2: Graph the Equations
#### Graphing [tex]\( y = -\frac{1}{5}x + 2 \)[/tex]
- Slope (m): [tex]\(-\frac{1}{5}\)[/tex]
- Y-intercept (b): [tex]\(2\)[/tex]
1. Start at the y-intercept (0, 2).
2. Plot another point using the slope. Since the slope is [tex]\(-\frac{1}{5}\)[/tex], go down 1 unit and right 5 units from the y-intercept. This gives the point (5, 1).
3. Draw the line through these two points.
#### Graphing [tex]\( y = -\frac{3}{5}x \)[/tex]
- Slope (m): [tex]\(-\frac{3}{5}\)[/tex]
- Y-intercept (b): [tex]\(0\)[/tex] (since there's no constant, the graph passes through the origin)
1. Start at the origin (0, 0).
2. Plot another point using the slope. Since the slope is [tex]\(-\frac{3}{5}\)[/tex], go down 3 units and right 5 units from the origin. This gives the point (5, -3).
3. Draw the line through these two points.
### Step 3: Determine the Solution
The solution to the system of equations is the point where the two lines intersect. Looking at the graphs of the equations:
1. The line [tex]\( y = -\frac{1}{5}x + 2 \)[/tex] crosses the line [tex]\( y = -\frac{3}{5}x \)[/tex] at a specific point.
2. To find the exact intersection point, we set the equations equal to each other and solve for [tex]\( x \)[/tex].
[tex]\[
-\frac{1}{5}x + 2 = -\frac{3}{5}x
\][/tex]
### Step 4: Solve the Equation
Add [tex]\(\frac{1}{5}x\)[/tex] to both sides:
[tex]\[
2 = -\frac{3}{5}x + \frac{1}{5}x
\][/tex]
[tex]\[
2 = -\frac{2}{5}x
\][/tex]
Multiply both sides by [tex]\(-\frac{5}{2}\)[/tex]:
[tex]\[
x = -5
\][/tex]
Now, substitute [tex]\( x = -5 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Using [tex]\( y = -\frac{3}{5}x \)[/tex]:
[tex]\[
y = -\frac{3}{5}(-5)
\][/tex]
[tex]\[
y = 3
\][/tex]
So, the solution to the system of equations is [tex]\( (x, y) = (-5, 3) \)[/tex].
### Conclusion
To summarize, the solution to the given system of equations is [tex]\( \boxed{(-5, 3)} \)[/tex], which represents the point of intersection on the graph.
