To solve the system of equations graphically, we need to first understand and plot each equation on the coordinate plane.
### 1. Analyzing and Plotting the Equations
#### Equation 1: [tex]\( y = 3x + 9 \)[/tex]
This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- The slope [tex]\( m \)[/tex] is 3.
- The y-intercept [tex]\( b \)[/tex] is 9.
To plot this line:
- Start at the y-intercept (0, 9).
- Use the slope [tex]\( 3 \)[/tex] to find another point. From (0,9), move up 3 units and right 1 unit to reach the point (1, 12).
#### Equation 2: [tex]\( 6x - 2y = 6 \)[/tex]
This is also a linear equation but in standard form [tex]\( Ax + By = C \)[/tex]. We can convert it to slope-intercept form to facilitate graphing.
First, solve for [tex]\( y \)[/tex]:
[tex]\[ 6x - 2y = 6 \][/tex]
Divide the entire equation by 2:
[tex]\[ 3x - y = 3 \][/tex]
Add [tex]\( y \)[/tex] to both sides and then subtract [tex]\( 3 \)[/tex] from both sides:
[tex]\[ y = 3x - 3 \][/tex]
Now, we have the slope-intercept form: [tex]\( y = 3x - 3 \)[/tex].
- The slope [tex]\( m \)[/tex] is 3.
- The y-intercept [tex]\( b \)[/tex] is -3.
To plot this line:
- Start at the y-intercept (0, -3).
- Use the slope [tex]\( 3 \)[/tex] to find another point. From (0, -3), move up 3 units and right 1 unit to reach the point (1, 0).
### 2. Graphing Both Lines
Graphing the two lines on the same coordinate plane:
1. Line 1 (Equation 1: [tex]\( y = 3x + 9 \)[/tex]):
- Plot (0, 9).
- Plot (1, 12).
2. Line 2 (Equation 2: [tex]\( y = 3x - 3 \)[/tex]):
- Plot (0, -3).
- Plot (1, 0).
### 3. Analyzing the Graph
Notice that both lines have the same slope (3) but different y-intercepts (9 and -3). Because the slopes are the same, the lines are parallel.
Parallel lines that are not coincident (i.e., not the same line) never intersect, which means there is no point that satisfies both equations simultaneously.
### Conclusion
Since the lines are parallel and will never meet, the system of equations has no solution.
Therefore, the correct option is:
- There is no solution.
