To graph the system of equations given by:
[tex]\[ 1.15x + 0.65y = 8.90 \][/tex]
[tex]\[ x - 3y = -15 \][/tex]
we need to follow these steps:
### 1. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] for each equation
#### Equation 1:
[tex]\[ 1.15x + 0.65y = 8.90 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ 0.65y = 8.90 - 1.15x \][/tex]
[tex]\[ y = \frac{8.90 - 1.15x}{0.65} \][/tex]
Simplify:
[tex]\[ y = \frac{8.90}{0.65} - \frac{1.15x}{0.65} \][/tex]
[tex]\[ y \approx 13.69 - 1.77x \][/tex]
#### Equation 2:
[tex]\[ x - 3y = -15 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ -3y = -15 - x \][/tex]
[tex]\[ y = \frac{-15 - x}{-3} \][/tex]
[tex]\[ y = 5 + \frac{x}{3} \][/tex]
### 2. Plot the two equations on a graph
- Equation 1: [tex]\( y \approx 13.69 - 1.77x \)[/tex]
- Equation 2: [tex]\( y = 5 + \frac{x}{3} \)[/tex]
We need to choose values of [tex]\( x \)[/tex] and find the corresponding [tex]\( y \)[/tex] values to plot both lines.
#### Plot points for Equation 1:
Using a few values of [tex]\( x \)[/tex]:
| [tex]\( x \)[/tex] | [tex]\( y \approx 13.69 - 1.77x \)[/tex] |
|--------|------------------------------|
| 0 | 13.69 |
| 5 | 4.84 |
| 10 | -4.01 |
#### Plot points for Equation 2:
Using a few values of [tex]\( x \)[/tex]:
| [tex]\( x \)[/tex] | [tex]\( y = 5 + \frac{x}{3} \)[/tex] |
|--------|--------------------------|
| 0 | 5 |
| 6 | 7 |
| 9 | 8 |
### 3. Finding the intersection point
The lines intersect where [tex]\( 1.15x + 0.65y = 8.90 \)[/tex] and [tex]\( x - 3y = -15 \)[/tex] are both satisfied. To find the exact intersection point, we solve the system of linear equations:
[tex]\[ 1.15x + 0.65y = 8.90 \][/tex]
[tex]\[ x - 3y = -15 \][/tex]
Using the method of substitution or elimination:
First, solve [tex]\( x - 3y = -15 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ x = -15 + 3y \][/tex]
Substitute [tex]\( x \)[/tex] into [tex]\( 1.15x + 0.65y = 8.90 \)[/tex]:
[tex]\[ 1.15(-15 + 3y) + 0.65y = 8.90 \][/tex]
[tex]\[ -17.25 + 3.45y + 0.65y = 8.90 \][/tex]
[tex]\[ -17.25 + 4.10y = 8.90 \][/tex]
[tex]\[ 4.10y = 8.90 + 17.25 \][/tex]
[tex]\[ 4.10y = 26.15 \][/tex]
[tex]\[ y = \frac{26.15}{4.10} \][/tex]
[tex]\[ y \approx 6.38 \][/tex]
Now substitute [tex]\( y \approx 6.38 \)[/tex] back into [tex]\( x = -15 + 3y \)[/tex]:
[tex]\[ x = -15 + 3(6.38) \][/tex]
[tex]\[ x = -15 + 19.14 \][/tex]
[tex]\[ x \approx 4.14 \][/tex]
So, the intersection point is:
[tex]\[ (x, y) \approx (4.14, 6.38) \][/tex]
### 4. Draw the graph
1. Draw the x-y axes.
2. Plot the points for each line and draw the lines through these points.
- For the first line [tex]\( y = 13.69 - 1.77x \)[/tex]:
- [tex]\( (0, 13.69) \)[/tex]
- [tex]\( (5, 4.84) \)[/tex]
- [tex]\( (10, -4.01) \)[/tex]
- For the second line [tex]\( y = 5 + \frac{x}{3} \)[/tex]:
- [tex]\( (0, 5) \)[/tex]
- [tex]\( (6, 7) \)[/tex]
- [tex]\( (9, 8) \)[/tex]
3. Mark the intersection point [tex]\((4.14, 6.38)\)[/tex] on the graph.
With these steps, you should be able to graph the system of equations and observe where they intersect.
