A Line Passes Through The Point \[$(10, 9)\$\] And Has A Slope Of \[$\frac{1}{2}\$\].Write An Equation In The Form \[$Ax + By = C\$\] For This Line. Use Integers For \[$A\$\], \[$B\$\], And \[$C\$\].

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Introduction

In mathematics, the equation of a line can be expressed in various forms, including the slope-intercept form, point-slope form, and standard form. The standard form of a line is given by the equation {Ax + By = C$}$, where {A$}$, {B$}$, and {C$}$ are integers. In this article, we will focus on finding the equation of a line that passes through a given point and has a specified slope.

The Given Information

We are given that a line passes through the point {(10, 9)$}$ and has a slope of {\frac{1}{2}$}$. Our goal is to find the equation of this line in the form {Ax + By = C$}$, where {A$}$, {B$}$, and {C$}$ are integers.

Using the Point-Slope Form

The point-slope form of a line is given by the equation {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 can substitute the given point {(10, 9)$}$ and the slope {\frac{1}{2}$}$ into the point-slope form to obtain:

{y - 9 = \frac{1}{2}(x - 10)$}$

Converting to Standard Form

To convert the point-slope form to the standard form {Ax + By = C$}$, we need to eliminate the fraction and simplify the equation. We can do this by multiplying both sides of the equation by 2, which is the denominator of the fraction:

${2(y - 9) = 2 \cdot \frac{1}{2}(x - 10)\$}

This simplifies to:

${2y - 18 = x - 10\$}

Rearranging the Terms

To put the equation in the standard form {Ax + By = C$}$, we need to rearrange the terms so that the x-term is on the left-hand side and the y-term is on the right-hand side. We can do this by adding 10 to both sides of the equation:

${2y - 18 + 10 = x - 10 + 10\$}

This simplifies to:

${2y - 8 = x\$}

Writing the Equation in Standard Form

To write the equation in the standard form {Ax + By = C$}$, we need to multiply both sides of the equation by -1 to make the coefficient of x positive:

{-x + 2y - 8 = 0$}$

This can be rewritten as:

{-x + 2y = 8$}$

Conclusion

In this article, we found the equation of a line that passes through the point {(10, 9)$}$ and has a slope of {\frac{1}{2}$}$. We used the point-slope form to derive the equation and then converted it to the standard form {Ax + By = C$}$. The final equation is {-x + 2y = 8$}$, where {A = -1$}$, {B = 2$}$, and {C = 8$}$.

Final Answer

The final answer is {-x + 2y = 8$}$.

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Introduction

In our previous article, we found the equation of a line that passes through the point {(10, 9)$}$ and has a slope of {\frac{1}{2}$}$. We used the point-slope form to derive the equation and then converted it to the standard form {Ax + By = C$}$. In this article, we will answer some common questions related to finding the equation of a line with a given point and slope.

Q&A

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 {(x_1, y_1)$}$ is a point on the line and {m$}$ is the slope.

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

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

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$}$, {B$}$, and {C$}$ are integers.

Q: How do I convert the point-slope form to the standard form?

A: To convert the point-slope form to the standard form, you need to eliminate the fraction and simplify the equation. You can do this by multiplying both sides of the equation by the denominator of the fraction.

Q: What is the final equation of the line that passes through the point {(10, 9)$}$ and has a slope of {\frac{1}{2}$}$?

A: The final equation of the line is {-x + 2y = 8$}$, where {A = -1$}$, {B = 2$}$, and {C = 8$}$.

Example Problems

Problem 1

Find the equation of a line that passes through the point {(4, 7)$}$ and has a slope of {\frac{3}{4}$}$.

Solution

Using the point-slope form, we can substitute the given point and the slope into the equation:

{y - 7 = \frac{3}{4}(x - 4)$}$

Multiplying both sides of the equation by 4, we get:

${4(y - 7) = 3(x - 4)\$}

Simplifying the equation, we get:

${4y - 28 = 3x - 12\$}

Rearranging the terms, we get:

${3x - 4y = -16\$}

Problem 2

Find the equation of a line that passes through the point {(2, 5)$}$ and has a slope of {\frac{2}{3}$}$.

Solution

Using the point-slope form, we can substitute the given point and the slope into the equation:

{y - 5 = \frac{2}{3}(x - 2)$}$

Multiplying both sides of the equation by 3, we get:

${3(y - 5) = 2(x - 2)\$}

Simplifying the equation, we get:

${3y - 15 = 2x - 4\$}

Rearranging the terms, we get:

${2x - 3y = -11\$}

Conclusion

In this article, we answered some common questions related to finding the equation of a line with a given point and slope. We also provided example problems to help illustrate the concepts. By following the steps outlined in this article, you should be able to find the equation of a line that passes through a given point and has a specified slope.

Final Answer

The final answer is {-x + 2y = 8$}$.