Question

which of the following is another pair of coordinates that is on the line that passes through (1,4) and is parallel to line g?
a. (4,5)
b. (5,5)
c. (4,6)

Explanation:

1. First, assume the slope - intercept form of a line is \(y = mx + b\). Since we don't know the slope \(m\) of line \(g\), but we know that parallel lines have the same slope. Let's find the slope between the given point \((1,4)\) and each option point using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

• For option A: Let \((x_1,y_1)=(1,4)\) and \((x_2,y_2)=(4,5)\).
• Calculate the slope \(m=\frac{5 - 4}{4 - 1}=\frac{1}{3}\).
• For option B: Let \((x_1,y_1)=(1,4)\) and \((x_2,y_2)=(5,5)\).
• Calculate the slope \(m=\frac{5 - 4}{5 - 1}=\frac{1}{4}\).
• For option C: Let \((x_1,y_1)=(1,4)\) and \((x_2,y_2)=(4,6)\).
• Calculate the slope \(m=\frac{6 - 4}{4 - 1}=\frac{2}{3}\).
• However, if we assume the line has a slope of \(\frac{2}{3}\) (by further context not given in the problem - but by the process of elimination or assuming a standard linear - relationship problem), we can also use the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(1,4)\) and \(m = \frac{2}{3}\).
• The point - slope form is \(y-4=\frac{2}{3}(x - 1)\).
• For option C, when \(x = 4\), \(y-4=\frac{2}{3}(4 - 1)\).
• \(y-4=\frac{2}{3}\times3\).
• \(y-4 = 2\), then \(y=6\).

Answer:

C. \((4,6)\)