Which Equation Represents A Line That Passes Through \[$(5,1)\$\] And Has A Slope Of \[$\frac{1}{2}\$\]?A. \[$y - 5 = \frac{1}{2}(x - 1)\$\]B. \[$y - \frac{1}{2} = 5(x - 1)\$\]C. \[$y - 1 = \frac{1}{2}(x -

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Which Equation Represents a Line That Passes Through a Given Point and Has a Specific Slope?

In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. When given a point through which the line passes and its slope, we can use the point-slope form to find the equation of the line. In this article, we will explore which equation represents a line that passes through the point {(5,1)$}$ and has a slope of {\frac{1}{2}$}$.

Understanding the Point-Slope Form

The point-slope form of a line is given by the equation:

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

where [(x1,y1)$]isapointonthelineand\[(x_1, y_1)\$] is a point on the line and \[m$] is the slope of the line. This form is useful when we are given a point and the slope of the line.

Given Information

We are given a point [(5,1)$]andtheslope\[(5,1)\$] and the slope \[\frac{1}{2}$]. We need to find the equation of the line that passes through this point and has this slope.

Option A: [$y - 5 = \frac{1}{2}(x - 1)$]

Let's analyze the first option:

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

We can rewrite this equation as:

[$y = \frac{1}{2}x + \frac{9}{2}$]

This is the slope-intercept form of a line, where the slope is [12$]andthey−interceptis\[\frac{1}{2}\$] and the y-intercept is \[\frac{9}{2}$]. However, we are given that the point [(5,1)$]liesontheline.Substituting\[(5,1)\$] lies on the line. Substituting \[x = 5$] and [$y = 1$] into the equation, we get:

[$1 = \frac{1}{2}(5) + \frac{9}{2}$]

Simplifying the equation, we get:

[$1 = \frac{5}{2} + \frac{9}{2}$]

[$1 = \frac{14}{2}$]

[$1 = 7$]

This is a contradiction, as [$1 \neq 7$]. Therefore, option A is not the correct equation.

Option B: [$y - \frac{1}{2} = 5(x - 1)$]

Let's analyze the second option:

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

We can rewrite this equation as:

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

This is not the slope-intercept form of a line, and it does not match the given slope of [$\frac{1}{2}$]. Therefore, option B is not the correct equation.

Option C: [$y - 1 = \frac{1}{2}(x - 5)$]

Let's analyze the third option:

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

We can rewrite this equation as:

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

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

This is the slope-intercept form of a line, where the slope is [12$]andthey−interceptis\[\frac{1}{2}\$] and the y-intercept is \[-\frac{3}{2}$]. We can substitute [x=5$]and\[x = 5\$] and \[y = 1$] into the equation to verify that the point [$(5,1)$] lies on the line:

[$1 = \frac{1}{2}(5) - \frac{3}{2}$]

Simplifying the equation, we get:

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

[$1 = \frac{2}{2}$]

[$1 = 1$]

This is true, as [$1 = 1$]. Therefore, option C is the correct equation.

In conclusion, the equation that represents a line that passes through the point [(5,1)$]andhasaslopeof\[(5,1)\$] and has a slope of \[\frac{1}{2}$] is:

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

This equation is in the point-slope form, where the point [(5,1)$]isgiven,andtheslopeis\[(5,1)\$] is given, and the slope is \[\frac{1}{2}$]. We can verify that this equation is true by substituting the point [$(5,1)$] into the equation.

The final answer is option C: [$y - 1 = \frac{1}{2}(x - 5)$].
Frequently Asked Questions (FAQs) About the Equation of a Line

In our previous article, we explored which equation represents a line that passes through the point [(5,1)$]andhasaslopeof\[(5,1)\$] and has a slope of \[\frac{1}{2}$]. We found that the correct equation is:

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

In this article, we will answer some frequently asked questions (FAQs) about the equation of a line.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by the equation:

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

where [(x1,y1)$]isapointonthelineand\[(x_1, y_1)\$] is a point on the line and \[m$] is the slope of the line.

Q: How do I find the equation of a line that passes through a given point and has a specific slope?

A: To find the equation of a line that passes through a given point and has a specific slope, you can use the point-slope form of a line. Simply substitute the given point and slope into the equation:

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

Q: What is the slope-intercept form of a line?

A: The slope-intercept form of a line is given by the equation:

[$y = mx + b$]

where [m$]istheslopeofthelineand\[m\$] is the slope of the line and \[b$] is the y-intercept.

Q: How do I convert the point-slope form of a line to the slope-intercept form?

A: To convert the point-slope form of a line to the slope-intercept form, you can simplify the equation by distributing the slope and combining like terms:

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

[$y - y_1 = mx - mx_1$]

[$y = mx - mx_1 + y_1$]

[$y = mx + (y_1 - mx_1)$]

Q: What is the standard form of a line?

A: The standard form of a line is given by the equation:

[$Ax + By = C$]

where [A$],\[A\$], \[B$], and [$C$] are constants.

Q: How do I convert the slope-intercept form of a line to the standard form?

A: To convert the slope-intercept form of a line to the standard form, you can multiply both sides of the equation by the denominator of the slope:

[$y = mx + b$]

[$y - b = mx$]

[$\frac{y - b}{m} = x$]

[$\frac{y - b}{m}x = 1$]

[$Ax + By = C$]

where [A=1m$],\[A = \frac{1}{m}\$], \[B = -1$], and [$C = b$].

In conclusion, we have answered some frequently asked questions (FAQs) about the equation of a line. We have discussed the point-slope form, slope-intercept form, and standard form of a line, and provided examples of how to convert between these forms.

The final answer is that the equation of a line can be represented in various forms, including the point-slope form, slope-intercept form, and standard form. By understanding these forms and how to convert between them, you can solve a wide range of problems involving the equation of a line.