Question

which equation represents the line that contains the points (2, 6)
\\( y = \frac{1}{2}x + 7 \\)
\\( y = \frac{1}{2}x + 5 \\)
\\( y = \frac{5}{2}x + 1 \\)
\\( y = \frac{5}{2}x + 11 \\)

Explanation:

Step1: Identify the problem type

This is a linear equation problem where we need to find the equation of a line passing through a given point (assuming the full problem has another point or slope, but here we can test the point (2,6) in each equation).

Step2: Test the point in each equation

• For \( y = \frac{1}{2}x + 7 \): Substitute \( x = 2 \), \( y=\frac{1}{2}(2)+7 = 1 + 7 = 8

eq 6 \).

• For \( y = \frac{1}{2}x + 5 \): Substitute \( x = 2 \), \( y=\frac{1}{2}(2)+5 = 1 + 5 = 6 \). This matches the y - value for \( x = 2 \).
• For \( y=\frac{5}{2}x + 1 \): Substitute \( x = 2 \), \( y=\frac{5}{2}(2)+1 = 5 + 1 = 6 \). Wait, let's check again. Wait, maybe there was a typo in the original problem (maybe the other point is missing). Wait, no, maybe I made a mistake. Wait, if the point is (2,6), let's check the third equation again. \( \frac{5}{2}\times2=5 \), \( 5 + 1 = 6 \). But wait, let's check the slope. Wait, maybe the original problem had two points. Let's assume that maybe the first point is (2,6) and another point, say, if we consider the slope. Wait, no, the user's problem might have a typo, but from the given options, let's re - check.

Wait, maybe the original problem has two points, like (2,6) and another point. But based on the given options, let's test (2,6) in each:

1. \( y=\frac{1}{2}x + 7 \): \( x = 2 \), \( y=\frac{1}{2}\times2+7=1 + 7 = 8

eq6 \)

1. \( y=\frac{1}{2}x + 5 \): \( x = 2 \), \( y=\frac{1}{2}\times2 + 5=1+5 = 6 \)
2. \( y=\frac{5}{2}x+1 \): \( x = 2 \), \( y=\frac{5}{2}\times2+1 = 5 + 1=6 \)
3. \( y=\frac{5}{2}x + 11 \): \( x = 2 \), \( y=\frac{5}{2}\times2+11=5 + 11 = 16

eq6 \)

Wait, now we have two equations that satisfy (2,6). But maybe the other point is, for example, if we consider the slope between two points. Wait, maybe the original problem was (2,6) and (0,5) (for the second equation, y - intercept 5) or (2,6) and (0,1) (for the third). But since the second equation gives \( y=\frac{1}{2}x + 5 \) which is a common form, and maybe the other point was (0,5). Let's assume that the correct equation is \( y=\frac{1}{2}x + 5 \) as it is a more likely linear equation with a smaller slope.

Answer:

\( y=\frac{1}{2}x + 5 \) (the option \( y=\frac{1}{2}x + 5 \))