Use the table to answer the question. x-value y-value if y=x+6 y-value if y=2x+3 0 6 3 1 7 5 2 8 7 3 9 9 4 10 11 A system of equations is displayed in the table. What is the solution for the system? (1 point)

Answer 1

To find the solution for the system of equations represented by the two functions \(y = x + 6\) and \(y = 2x + 3\), we can compare the y-values in the table for each corresponding x-value.

1. Start with the equations:

   • Equation 1: \(y = x + 6\)
   • Equation 2: \(y = 2x + 3\)

2. Check the given x-values and calculate the corresponding y-values for both equations:

   • For \(x = 0\):
     • \(y = 0 + 6 = 6\) (Equation 1)
     • \(y = 2(0) + 3 = 3\) (Equation 2)
   • For \(x = 1\):
     • \(y = 1 + 6 = 7\) (Equation 1)
     • \(y = 2(1) + 3 = 5\) (Equation 2)
   • For \(x = 2\):
     • \(y = 2 + 6 = 8\) (Equation 1)
     • \(y = 2(2) + 3 = 7\) (Equation 2)
   • For \(x = 3\):
     • \(y = 3 + 6 = 9\) (Equation 1)
     • \(y = 2(3) + 3 = 9\) (Equation 2)
   • For \(x = 4\):
     • \(y = 4 + 6 = 10\) (Equation 1)
     • \(y = 2(4) + 3 = 11\) (Equation 2)

3. Now we can find the x-value for which the y-values from both equations are equal. From our calculations above, the only point where the y-values from both equations match is:

   • When \(x = 3\):
     • \(y = 9\) from both equations

4. Therefore, the solution to the system of equations is: \[ (x, y) = (3, 9) \]

So the final answer is: \[ \text{The solution for the system is } (3, 9). \]

Answer 2

How many solutions does this system of equations have? (1 point)

Answer 3

To determine how many solutions the system of equations has, we need to analyze the equations:

1. Equation 1: \(y = x + 6\)
2. Equation 2: \(y = 2x + 3\)

From our previous analysis, we found that these two equations intersect at one point: \((3, 9)\).

To further analyze, let's consider the characteristics of the two equations:

• Equation 1 (\(y = x + 6\)) is a linear equation with a slope of 1.
• Equation 2 (\(y = 2x + 3\)) is also a linear equation but with a slope of 2.

Since the slopes of the two lines are different, they will intersect at exactly one point. Therefore, this system of equations has exactly one solution.

So the answer is: \[ \text{The system has 1 solution.} \]

Answer 4

Does the system of equations y=1/2x−3 and y=1/2x+2 describe a zero solution system or an infinite solution system? Enter 1 for zero or 2 for infinite.(1 point)

Answer 5

To determine whether the system of equations has zero solutions or infinite solutions, we should analyze the given equations:

1. Equation 1: \(y = \frac{1}{2}x - 3\)
2. Equation 2: \(y = \frac{1}{2}x + 2\)

Both equations are in the form \(y = mx + b\):

• Both have the same slope \(m = \frac{1}{2}\).
• However, they have different y-intercepts: \(-3\) for the first equation and \(2\) for the second equation.

When two linear equations have the same slope but different y-intercepts, it indicates that the lines are parallel. Parallel lines do not intersect, which means there are no points that satisfy both equations simultaneously.

Thus, the system of equations has zero solutions.

Based on the formatting of your question, the answer is: \[ \text{1} \text{ (for zero solutions).} \]