Question

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

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

Answer 2

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 3

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 4

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 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:
\[
\text{1} \text{ (for zero solutions).}
\]