Questions: The table shows four systems of linear equations and their respective graphs. Determine the number of solutions to each system of equations. System of Equations Graph Number of Solutions --------- 2y - x = 6 y = 2x + 7 x + 4y = 16 y = -1/4 x + 2 y = -1/2 x + 3 y = -1/3 x + 2 y = 1/2 x + 1 2y - x = 2 no solution one solution infinitely many solutions

To determine the number of solutions for each system of linear equations, we can analyze the equations to see if they are parallel (no solution), intersect at one point (one solution), or are the same line (infinitely many solutions). This can be done by comparing the slopes and intercepts of the lines.

1. For the first system, compare the slopes of the two lines. If they are equal and the intercepts are different, the lines are parallel and there is no solution. If the slopes are different, there is one solution.
2. For the second system, again compare the slopes. If they are equal and the intercepts are different, the lines are parallel and there is no solution. If the slopes are different, there is one solution.
3. For the third system, compare the slopes. If they are equal and the intercepts are different, the lines are parallel and there is no solution. If the slopes are different, there is one solution.

Das erste Gleichungssystem besteht aus den Gleichungen: \[ 2y - x = 6 \] \[ y = 2x + 7 \]

Um die Anzahl der Lösungen zu bestimmen, setzen wir die zweite Gleichung in die erste ein: \[ 2(2x + 7) - x = 6 \] \[ 4x + 14 - x = 6 \] \[ 3x = -8 \] \[ x = -\frac{8}{3} \]

Setzen wir \( x = -\frac{8}{3} \) in die zweite Gleichung ein: \[ y = 2\left(-\frac{8}{3}\right) + 7 = \frac{5}{3} \]

Das System hat eine eindeutige Lösung: \((x, y) = \left(-\frac{8}{3}, \frac{5}{3}\right)\).

Das zweite Gleichungssystem besteht aus den Gleichungen: \[ x + 4y = 16 \] \[ y = 2 - \frac{1}{4}x \]

Setzen wir die zweite Gleichung in die erste ein: \[ x + 4\left(2 - \frac{1}{4}x\right) = 16 \] \[ x + 8 - x = 16 \] \[ 8 = 16 \]

Da diese Gleichung falsch ist, gibt es keine Lösung für dieses System. Die Geraden sind parallel.

Das dritte Gleichungssystem besteht aus den Gleichungen: \[ y = -\frac{1}{2}x + 3 \] \[ y = -\frac{1}{3}x + 2 \]

Setzen wir die beiden Gleichungen gleich: \[ -\frac{1}{2}x + 3 = -\frac{1}{3}x + 2 \] \[ -\frac{1}{2}x + \frac{1}{3}x = 2 - 3 \] \[ -\frac{1}{6}x = -1 \] \[ x = 6 \]

Setzen wir \( x = 6 \) in eine der beiden Gleichungen ein: \[ y = -\frac{1}{2}(6) + 3 = 0 \]

Das System hat eine eindeutige Lösung: \((x, y) = (6, 0)\).

• Für das erste System: \(\boxed{(x, y) = \left(-\frac{8}{3}, \frac{5}{3}\right)}\)
• Für das zweite System: \(\boxed{\text{keine Lösung}}\)
• Für das dritte System: \(\boxed{(x, y) = (6, 0)}\)