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 \( n\):
- 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.
5 answers