To solve the system of equations graphically, we'll plot both the equations on the same set of axes and find their point of intersection. The system of equations is given by:
1. [tex]\( y = \frac{1}{6} x + 5 \)[/tex]
2. [tex]\( 4x + 3y = -12 \)[/tex]
Step-by-Step Solution:
### Equation 1: [tex]\( y = \frac{1}{6} x + 5 \)[/tex]
This is a linear equation in the slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Slope ([tex]\( m \)[/tex]) = [tex]\(\frac{1}{6}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = 5
#### Plotting Equation 1
1. Start by plotting the y-intercept: [tex]\( (0,5) \)[/tex].
2. Use the slope to find another point. A slope of [tex]\(\frac{1}{6}\)[/tex] means for each increase of 6 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1 unit.
- For example, if [tex]\( x = 6 \)[/tex]:
[tex]\[
y = \frac{1}{6}(6) + 5 = 1 + 5 = 6
\][/tex]
- Another point is [tex]\( (6, 6) \)[/tex].
3. Plot these two points: [tex]\( (0, 5) \)[/tex] and [tex]\( (6, 6) \)[/tex].
4. Draw a line through them to represent the equation [tex]\( y = \frac{1}{6} x + 5 \)[/tex].
### Equation 2: [tex]\( 4x + 3y = -12 \)[/tex]
To plot this equation, we'll convert it to the slope-intercept form [tex]\( y = mx + b \)[/tex].
#### Solving for y:
[tex]\[
3y = -12 - 4x \implies y = -\frac{4}{3}x - 4
\][/tex]
- Slope ([tex]\( m \)[/tex]) = [tex]\(-\frac{4}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = [tex]\(-4\)[/tex]
#### Plotting Equation 2
1. Start by plotting the y-intercept: [tex]\( (0, -4) \)[/tex].
2. Use the slope to find another point. A slope of [tex]\(-\frac{4}{3}\)[/tex] means for each increase of 3 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 4 units.
- For example, if [tex]\( x = 3 \)[/tex]:
[tex]\[
y = -\frac{4}{3}(3) - 4 = -4 - 4 = -8
\][/tex]
- Another point is [tex]\( (3, -8) \)[/tex].
3. Plot these two points: [tex]\( (0, -4) \)[/tex] and [tex]\( (3, -8) \)[/tex].
4. Draw a line through them to represent the equation [tex]\( y = -\frac{4}{3} x - 4 \)[/tex].
### Finding the Point of Intersection
By plotting both lines on the same set of axes, the point where they intersect represents the solution to the system of equations.
The lines will intersect at approximately:
[tex]\(( -15 , 2 )\)[/tex]
So the solution to the system of equations [tex]\( y = \frac{1}{6} x + 5 \)[/tex] and [tex]\( 4x + 3y = -12 \)[/tex] is [tex]\( x = -15 \)[/tex], [tex]\( y = 2 \)[/tex].
