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

Question

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:

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