Question

Which equation represents a line that passes through (5, 1) and has a slope of StartFraction one-half EndFraction?

y – 5 = y minus 5 equals StartFraction one-half EndFraction left-parenthesis x minus 1 right-parenthesis.(x –1)
y – y minus StartFraction one-half EndFraction equals 5 left-parenthesis x minus 1 right-parenthesis. = 5(x –1)
y – 1 = y minus 1 equals StartFraction one-half EndFraction left-parenthesis x minus 5 right-parenthesis.(x –5)
y – 1 = 5y minus 1 equals 5 left-parenthesis x minus StartFraction one-half EndFraction right-parenthesis.

Answer 1

To find the equation of a line that passes through the point (5, 1) with a slope of \( \frac{1}{2} \), we can use the point-slope form of the line equation, which is given by:

\[
y - y_1 = m(x - x_1)
\]

where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. In this case, we have:

- \( (x_1, y_1) = (5, 1) \)
- \( m = \frac{1}{2} \)

Substituting these values into the point-slope form:

\[
y - 1 = \frac{1}{2}(x - 5)
\]

This simplifies to:

\[
y - 1 = \frac{1}{2}x - \frac{5}{2}
\]

To further arrange this into a typical line equation form, we can add 1 to both sides:

\[
y = \frac{1}{2}x - \frac{5}{2} + 1
\]

Combine the constant terms:

\[
-\frac{5}{2} + 1 = -\frac{5}{2} + \frac{2}{2} = -\frac{3}{2}
\]

So we have:

\[
y = \frac{1}{2}x - \frac{3}{2}
\]

Now checking the original options, we want to find one that matches \( y - 1 = \frac{1}{2}(x - 5) \).

The correct equation from your provided options is:

- \( y - 1 = \frac{1}{2}(x - 5) \)

So the answer is:

\[
y - 1 = \frac{1}{2}(x - 5)
\]