Question

10. A system of equations is displayed in the table. What is the solution of the system?
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

Answer 1

shorter please

Answer 2

To find the solution of the system of equations based on the table, we need to identify the x-value where the values of \(y\) from both equations are equal. The equations given are:

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

We'll compare the \(y\)-values for each \(x\) from the table and see where they match:

For \(x = 0\):
- \(y = x + 6\) gives \(y = 0 + 6 = 6\)
- \(y = 2x + 3\) gives \(y = 2(0) + 3 = 3\)

For \(x = 1\):
- \(y = x + 6\) gives \(y = 1 + 6 = 7\)
- \(y = 2x + 3\) gives \(y = 2(1) + 3 = 5\)

For \(x = 2\):
- \(y = x + 6\) gives \(y = 2 + 6 = 8\)
- \(y = 2x + 3\) gives \(y = 2(2) + 3 = 7\)

For \(x = 3\):
- \(y = x + 6\) gives \(y = 3 + 6 = 9\)
- \(y = 2x + 3\) gives \(y = 2(3) + 3 = 9\) (Equal values)

For \(x = 4\):
- \(y = x + 6\) gives \(y = 4 + 6 = 10\)
- \(y = 2x + 3\) gives \(y = 2(4) + 3 = 11\)

The only point where the \(y\) values are equal is at \(x = 3\), where both equations give \(y = 9\).

Therefore, the solution to the system of equations is:

\[
\boxed{(3, 9)}
\]

Answer 3

The solution to the system of equations can be found by comparing the \(y\)-values for each \(x\):

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

By checking the values in the table:

- For \(x = 3\):
- \(y = 3 + 6 = 9\) (from \(y = x + 6\))
- \(y = 2(3) + 3 = 9\) (from \(y = 2x + 3\))

Both equations yield \(y = 9\) at \(x = 3\).

Thus, the solution is:

\[
\boxed{(3, 9)}
\]