Answer:
Let's analyze each system of equations and classify them based on the number of solutions they have:
1) y = 11 − 2x
4x − y = 7
This system of equations represents two lines. The first equation is in slope-intercept form, and the second equation is in standard form. Since the equations have different slopes and different y-intercepts, they intersect at a single point. Thus, the system has a single solution.
2) x = 12 − 3y
3x + 9y = 24
The first equation represents a line, and the second equation is a linear equation. Since the first equation can be rewritten as 3y = 12 - x or y = 4 - (1/3)x, it indicates that the slope-intercept form can't be satisfied. Both equations are equivalent and represent the same line. Therefore, the system has infinitely many solutions.
3) 2x + y = 7
-4x = 2y + 14
The first equation represents a line, and the second equation is also a linear equation. If we simplify the second equation, we get y = -2x - 7, which is equivalent to the first equation. Thus, the system has infinitely many solutions.
4) x + y = 15
2x − y = 15
Both equations are in standard form. By adding the equations, we eliminate y and get 3x = 30, which simplifies to x = 10. Substituting x = 10 into either equation, we find y = 5. Therefore, the system has a single solution.
5) x + 4y = 6
2x = 12 − 8y
The first equation represents a line, and the second equation is a linear equation. By simplifying the second equation, we get x = 6 - 8y, which is equivalent to the first equation. Therefore, the system has infinitely many solutions.
To summarize:
- System 1: Single solution.
- System 2: Infinitely many solutions.
- System 3: Infinitely many solutions.
- System 4: Single solution.
- System 5: Infinitely many solutions.