rewrite the equation 3x + y = 7 in slope-intercept form to select the statement that correctly describes the system for equations: 3x + y = 7, y = -3x + 7

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 no infinite number of solutions because the graphs of the two equations occupy the same line
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 (which is of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept), we can isolate \(y\):

Starting with the original equation:

\[ 3x + y = 7 \]

Subtract \(3x\) from both sides:

\[ y = -3x + 7 \]

This is now in slope-intercept form, where the slope \(m = -3\) and the y-intercept \(b = 7\).

Now, comparing this with the second equation \(y = -3x + 7\), we can see that both equations represent the same line.

Thus, we can conclude:

• The equation \(3x + y = 7\) is \(y = -3x + 7\) in slope-intercept form, which means that statement #2 describes the system of equations: "the system of equations has an infinite number of solutions because the graphs of the two equations occupy the same line."

So the final answer is:

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