Answer :
To solve the system of equations by graphing, follow these steps:
The system of equations is:
[tex]\[ \left\{\begin{array}{l}
-6x - y = -9 \\
\frac{1}{3} y = 3 - 2x
\end{array}\right. \][/tex]
First, let's rewrite both equations 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.
1. For the first equation [tex]\( -6x - y = -9 \)[/tex]:
- Isolate [tex]\( y \)[/tex]:
[tex]\[ -6x - y = -9 \][/tex]
[tex]\[ y = -6x + 9 \][/tex]
2. For the second equation [tex]\( \frac{1}{3} y = 3 - 2x \)[/tex]:
- Multiply both sides by 3 to clear the fraction:
[tex]\[ y = 9 - 6x \][/tex]
Now, we have the equations in slope-intercept form:
[tex]\[ y = -6x + 9 \][/tex]
[tex]\[ y = 9 - 6x \][/tex]
Upon simplification, we observe that both equations have the same line:
[tex]\[ y = -6x + 9 \][/tex]
Therefore, both lines are actually the same line, which means they overlap completely.
Since the lines are identical, there are infinitely many solutions, as every point on the line is a solution to the system. Hence, the system does not have a single unique solution but rather has infinitely many solutions, all the points on the line [tex]\( y = -6x + 9 \)[/tex].
The system of equations is:
[tex]\[ \left\{\begin{array}{l}
-6x - y = -9 \\
\frac{1}{3} y = 3 - 2x
\end{array}\right. \][/tex]
First, let's rewrite both equations 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.
1. For the first equation [tex]\( -6x - y = -9 \)[/tex]:
- Isolate [tex]\( y \)[/tex]:
[tex]\[ -6x - y = -9 \][/tex]
[tex]\[ y = -6x + 9 \][/tex]
2. For the second equation [tex]\( \frac{1}{3} y = 3 - 2x \)[/tex]:
- Multiply both sides by 3 to clear the fraction:
[tex]\[ y = 9 - 6x \][/tex]
Now, we have the equations in slope-intercept form:
[tex]\[ y = -6x + 9 \][/tex]
[tex]\[ y = 9 - 6x \][/tex]
Upon simplification, we observe that both equations have the same line:
[tex]\[ y = -6x + 9 \][/tex]
Therefore, both lines are actually the same line, which means they overlap completely.
Since the lines are identical, there are infinitely many solutions, as every point on the line is a solution to the system. Hence, the system does not have a single unique solution but rather has infinitely many solutions, all the points on the line [tex]\( y = -6x + 9 \)[/tex].
