Linear Algebra Basics: Systems. Game on.

Rewrite in slope-intercept form: $y = -2x + 7$ and $y = x - 2$. The slopes $-2$ and $1$ are different, so the lines intersect at exactly one point.

Rewrite in slope-intercept form: $y = \dfrac{5}{3}x - \dfrac{4}{3}$ and $y = -2x + 7$. The slopes $\dfrac{5}{3}$ and $-2$ are different, so the lines intersect at exactly one point.

Rewrite in slope-intercept form: $y = \dfrac{1}{2}x - \dfrac{3}{2}$ and $y = \dfrac{1}{2}x - 2$. The slopes are the same, but the $y$-intercepts differ, so the lines are parallel and the system has zero solutions.

Rewrite in slope-intercept form: $y = -\dfrac{1}{3}x + \dfrac{4}{3}$ and $y = -\dfrac{1}{3}x + \dfrac{4}{3}$. The slopes and $y$-intercepts match, so the equations describe the same line and the system has infinitely many solutions.

Rewrite in slope-intercept form: $y = -\dfrac{3}{4}x + \dfrac{5}{2}$ and $y = \dfrac{1}{2}x - \dfrac{1}{2}$. The slopes $-\dfrac{3}{4}$ and $\dfrac{1}{2}$ are different, so the lines intersect at exactly one point.

Rewrite in slope-intercept form: $y = -\dfrac{2}{3}x + \dfrac{4}{3}$ and $y = -\dfrac{2}{3}x + \dfrac{7}{3}$. The slopes are the same, but the $y$-intercepts differ, so the lines are parallel and the system has zero solutions.

Rewrite in slope-intercept form: $y = 2x - 3$ and $y = 2x - 3$. The slopes and $y$-intercepts match, so the equations describe the same line and the system has infinitely many solutions.

Rewrite in slope-intercept form: $y = -x + 5$ and $y = 3x - 3$. The slopes $-1$ and $3$ are different, so the lines intersect at exactly one point.