Question

what is the solution to the system of equations?
$2x - y = 7$
$y = 2x + 3$
(2, 3) (2, 7) infinite number of solutions no solution

Explanation:

Step1: Rewrite the first equation in slope-intercept form.

Starting with \(2x - y = 7\), we can solve for \(y\):
\( -y = -2x + 7\)
Multiply both sides by -1: \(y = 2x - 7\)

Step2: Compare the slopes and y-intercepts.

The second equation is \(y = 2x + 3\).
For a linear equation in the form \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept.

• The slope (\(m\)) of \(y = 2x - 7\) is \(2\).
• The slope (\(m\)) of \(y = 2x + 3\) is \(2\).
• The y-intercept (\(b\)) of \(y = 2x - 7\) is \(-7\).
• The y-intercept (\(b\)) of \(y = 2x + 3\) is \(3\).

Since the two lines have the same slope (\(m = 2\)) but different y-intercepts (\(-7\) vs. \(3\)), they are parallel lines. Parallel lines never intersect, so there is no solution to the system.

Answer:

no solution