Question

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

Answer 1

To rewrite the equation \(3x + y = 7\) in slope-intercept form (which is \(y = mx + b\)), we isolate \(y\) on one side of the equation:

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

Now we have the equation in slope-intercept form:
\[y = -3x + 7\]

Now, we can analyze the system of equations:

1. The first equation is \(3x + y = 7\), which we rewrote as \(y = -3x + 7\).
2. The second equation is also \(y = -3x + 7\).

Since both equations are identical, they represent the same line.

Thus, we can conclude:

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

is the correct statement.