To graph the given system of equations, follow these steps:
System of Equations:
1. [tex]\( 1.15x + 0.65y = 8.90 \)[/tex]
2. [tex]\( x - 3y = -15 \)[/tex]
First, let's find the slope-intercept form ( [tex]\( y = mx + b \)[/tex] ) for both equations.
### Step-by-Step Solution:
1. Rearrange the first equation [tex]\( 1.15x + 0.65y = 8.90 \)[/tex] to slope-intercept form:
[tex]\[ 1.15x + 0.65y = 8.90 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 0.65y = 8.90 - 1.15x \][/tex]
Divide both sides by [tex]\( 0.65 \)[/tex]:
[tex]\[ y = \frac{8.90 - 1.15x}{0.65} \][/tex]
Simplify:
[tex]\[ y = \frac{8.90}{0.65} - \frac{1.15}{0.65}x \][/tex]
[tex]\[ y = 13.69 - 1.77x \][/tex]
So, the first equation in slope-intercept form is:
[tex]\[ y = -1.77x + 13.69 \][/tex]
2. Rearrange the second equation [tex]\( x - 3y = -15 \)[/tex] to slope-intercept form:
[tex]\[ x - 3y = -15 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ -3y = -15 - x \][/tex]
Divide both sides by [tex]\( -3 \)[/tex]:
[tex]\[ y = \frac{-15 - x}{-3} \][/tex]
Simplify:
[tex]\[ y = 5 + \frac{1}{3}x \][/tex]
So, the second equation in slope-intercept form is:
[tex]\[ y = \frac{1}{3}x + 5 \][/tex]
### Plotting the Equations:
Now, we will plot these equations on a graph.
1. First Equation: [tex]\( y = -1.77x + 13.69 \)[/tex]
- y-intercept (b): 13.69 (the point where the line crosses the y-axis, set [tex]\( x = 0 \)[/tex] and find [tex]\( y \)[/tex])
- Slope (m): -1.77 (for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1.77 units)
2. Second Equation: [tex]\( y = \frac{1}{3}x + 5 \)[/tex]
- y-intercept (b): 5 (the point where the line crosses the y-axis, set [tex]\( x = 0 \)[/tex] and find [tex]\( y \)[/tex])
- Slope (m): [tex]\(\frac{1}{3}\)[/tex] (for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\(\frac{1}{3}\)[/tex] units)
Steps to plot:
1. Draw the x-axis and y-axis.
2. Mark the y-intercepts for both equations:
- First equation: [tex]\( y = 13.69 \)[/tex]
- Second equation: [tex]\( y = 5 \)[/tex]
3. Use the slopes to find another point on each line:
- For [tex]\( y = -1.77x + 13.69 \)[/tex], from the intercept (0, 13.69), go down 1.77 units and right 1 unit to find another point.
- For [tex]\( y = \frac{1}{3}x + 5 \)[/tex], from the intercept (0, 5), go up [tex]\(\frac{1}{3}\)[/tex] units and right 1 unit to find another point.
4. Draw straight lines through these points.
Interpreting the Graph:
The intersection point of the two lines represents the solution to the system of equations. This is the point where both equations are satisfied simultaneously.
By graphing [tex]\( y = -1.77x + 13.69 \)[/tex] and [tex]\( y = \(\frac{1}{3}\)[/tex] x + 5 \), you will see the point where they intersect. This is the solution to the system.
