To determine the solution to the system of equations by graphing, let's follow these steps:
1. Get the Equations in the Same Form:
The given equations are:
[tex]\[
y = -3x + 2
\][/tex]
[tex]\[
-\frac{1}{3} y = x - \frac{2}{3}
\][/tex]
First, let's convert the second equation into the form [tex]\( y = mx + b \)[/tex]:
[tex]\[
-\frac{1}{3} y = x - \frac{2}{3}
\][/tex]
Multiply both sides by [tex]\(-3\)[/tex] to clear the fraction:
[tex]\[
y = -3x + 2
\][/tex]
2. Compare the Equations:
Now we have:
[tex]\[
y = -3x + 2
\][/tex]
[tex]\[
y = -3x + 2
\][/tex]
Both equations are exactly the same. This means that the two lines represented by these equations are the same line.
3. Determine the Nature of the Solutions:
Since the two lines coincide, every point on the line is a solution. This is not a case of one unique solution nor no solution but actually infinite solutions, specifically all the points that lie on the line [tex]\( y = -3x + 2 \)[/tex].
4. Identify the Correct Answer:
While graphing the equations, we can see that all points on the line [tex]\( y = -3x + 2 \)[/tex] are solutions. Thus, there are infinitely many solutions to this system.
Given the choices:
- One solution: [tex]\(\{0, 2\}\)[/tex]
- No solution: [tex]\(\{\}\)[/tex]
- Solutions: all numbers on the line
- One solution: [tex]\(\{6, 0\}\)[/tex]
The correct answer is:
[tex]\[
\text{Solutions: all numbers on the line}
\][/tex]
