Question

Line n is shown on the coordinate grid. The table of ordered pairs represents some points on line p.



Line p
x
y
−1
6
0
4
1
2
2
0

Which system of equations represents line n and line p?
A.
3x−y=2
2x+y=4

B.
3x+y=−2
2x+y=4

C.
3x−y=2
x+2y=4

D.
3x+y=−2
x+2y=4

Answer 1

To determine which system of equations represents lines \( n \) and \( p \), we need to find the equations that correspond to the given points for line \( p \).

### Points for Line \( p \)
The table gives us the following points for line \( p \):
- \((-1, 6)\)
- \((0, 4)\)
- \((1, 2)\)
- \((2, 0)\)

### Finding the Equation for Line \( p \)
Let's start by calculating the slope (\(m\)) of line \( p\) using the points \((0, 4)\) and \((1, 2)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{1 - 0} = \frac{-2}{1} = -2
\]
Now that we have the slope, we can use the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\). Using the point \((0,4)\):
\[
y - 4 = -2(x - 0) \implies y - 4 = -2x \implies y = -2x + 4
\]
This can be rewritten in standard form:
\[
2x + y = 4
\]

### Checking the Options
Now, let's examine the options to see which contains \(2x + y = 4\):

A. \(3x - y = 2\) and \(2x + y = 4\) → Corresponds to line \(p\).

B. \(3x + y = -2\) and \(2x + y = 4\) → Corresponds to line \(p\).

C. \(3x - y = 2\) and \(x + 2y = 4\) → \(x + 2y = 4\) is not \(2x + y = 4\).

D. \(3x + y = -2\) and \(x + 2y = 4\) → \(x + 2y = 4\) is not \(2x + y = 4\).

### Conclusion
Now we need to check what could be the equation representing line \( n \). However, based on the calculations above, so far \(2x + y = 4\) is confirmed for line \( p \).

Only option **A** contains the equation for line \( p \) as well as a second equation that needs to be checked for line \( n\):

1. **For** \(3x - y = 2\):
- You can rewrite this as \(y = 3x - 2\).
- If you test the point from line \(p\), it does not satisfy \(3x - y = 2\) for our computed candidates from \( p\).

Therefore the correct system based on what we can confirm so far is:
**A**. \(3x−y=2\) and \(2x+y=4\).

If further information / coordinates for line \( n\) is provided, we can accurately define it, but based on the current information, **A** is the right set for line **p**.