Rewrite the equation 3x+y=7 in slope-intercept form to select the statement that correctly describes the system of equations

Question

Statement #1: The system of equations has no solutions because the graphs of the two equations are parallel lines.

Statement #2: The system of equations has an infinite number of solutions because the graphs of the two equations occupy the same line.

(2 points)
The equation 3x+y=7 is y=  in slope intercept form, which means that statement # describes the system of equations

Answer 1

To rewrite the equation \(3x + y = 7\) in slope-intercept form (y = mx + b), we need to solve for \(y\):

1. Start with the original equation: \[ 3x + y = 7 \]
2. Subtract \(3x\) from both sides to isolate \(y\): \[ y = -3x + 7 \]

Now we have the equation in slope-intercept form: \(y = -3x + 7\).

Analyzing Statements

1. Parallel Lines: For two lines to be parallel, they must have the same slope but different y-intercepts. This would imply that there are no solutions because they will never intersect.

2. Same Line: For two lines to be the same (coincident), they must have the same slope and the same y-intercept. This would imply that there are an infinite number of solutions since they overlap entirely.

Without the second equation to compare against \(y = -3x + 7\), we cannot definitively assign the correct statement about the system of equations.

If the second equation also has the form \(y = -3x + b\) (where \(b \neq 7\)), then it is Statement #1 (no solutions).

If the second equation is \(y = -3x + 7\), then it is Statement #2 (infinite solutions).

Since you have only provided one equation, you should complete your analysis with the second equation to choose between the two statements correctly.

Final Answer

The equation \(3x + y = 7\) is \(y = -3x + 7\) in slope-intercept form, which means that statement # cannot be determined describes the system of equations without additional information about the second equation.