Use The Point-slope Equation To Identify The Slope And The Coordinates Of A Point On The Line $y - 4 = \frac{1}{2}(x - 1$\].The Slope Of The Line Is $\square$A Point On The Line Is $\square$

Introduction

The point-slope equation is a fundamental concept in mathematics, particularly in algebra and geometry. It provides a powerful tool for identifying the slope and coordinates of a point on a line. In this article, we will delve into the world of point-slope equations and explore how to use them to uncover the slope and a point on the line $y - 4 = \frac{1}{2}(x - 1)$.

Understanding the Point-Slope Equation

The point-slope equation is a linear equation that takes the form $y - y_1 = m(x - x_1)$, where $m$ is the slope of the line, and $(x_1, y_1)$ is a point on the line. The equation is derived from the slope formula, which is $m = \frac{y_2 - y_1}{x_2 - x_1}$. By rearranging the slope formula, we get the point-slope equation.

Solving the Given Equation

Now, let's apply the point-slope equation to the given equation $y - 4 = \frac{1}{2}(x - 1)$. To do this, we need to rewrite the equation in the standard form of the point-slope equation, which is $y - y_1 = m(x - x_1)$.

Step 1: Rewrite the Equation

First, we need to rewrite the equation by isolating the $y$ term on the left-hand side. We can do this by adding $4$ to both sides of the equation:

$$y - 4 + 4 = \frac{1}{2}(x - 1) + 4$$

This simplifies to:

$$y = \frac{1}{2}(x - 1) + 4$$

Step 2: Identify the Slope

Now that we have the equation in the standard form, we can identify the slope $m$. In this case, the slope is $\frac{1}{2}$.

Step 3: Identify a Point on the Line

To identify a point on the line, we need to find the coordinates $(x_1, y_1)$. We can do this by substituting a value for $x$ into the equation and solving for $y$. Let's choose $x = 1$:

$$y = \frac{1}{2}(1 - 1) + 4$$

This simplifies to:

$$y = 4$$

So, a point on the line is $(1, 4)$.

Conclusion

In this article, we used the point-slope equation to identify the slope and coordinates of a point on the line $y - 4 = \frac{1}{2}(x - 1)$. We rewrote the equation in the standard form of the point-slope equation, identified the slope, and found a point on the line. The point-slope equation is a powerful tool for solving linear equations and is an essential concept in mathematics.

Key Takeaways

• The point-slope equation is a linear equation that takes the form $y - y_1 = m(x - x_1)$.
• The slope $m$ is the coefficient of the $x$ term.
• A point on the line is given by the coordinates $(x_1, y_1)$.
• The point-slope equation can be used to solve linear equations and find the slope and coordinates of a point on the line.

Further Reading

If you want to learn more about the point-slope equation and its applications, I recommend checking out the following resources:

• Khan Academy: Point-Slope Form
• Mathway: Point-Slope Form
• Wolfram Alpha: Point-Slope Form

References

• [1] Larson, R. (2019). Algebra and Trigonometry. Cengage Learning.
• [2] Sullivan, M. (2018). Algebra and Trigonometry. Pearson Education.

Q: What is the point-slope equation?

A: The point-slope equation is a linear equation that takes the form $y - y_1 = m(x - x_1)$, where $m$ is the slope of the line, and $(x_1, y_1)$ is a point on the line.

Q: How do I rewrite the point-slope equation in the standard form?

A: To rewrite the point-slope equation in the standard form, you need to isolate the $y$ term on the left-hand side. You can do this by adding or subtracting the same value to both sides of the equation.

Q: How do I identify the slope in the point-slope equation?

A: The slope $m$ is the coefficient of the $x$ term in the point-slope equation.

Q: How do I identify a point on the line in the point-slope equation?

A: To identify a point on the line, you need to find the coordinates $(x_1, y_1)$. You can do this by substituting a value for $x$ into the equation and solving for $y$.

Q: Can I use the point-slope equation to solve linear equations?

A: Yes, you can use the point-slope equation to solve linear equations. By rewriting the equation in the standard form, you can identify the slope and a point on the line.

Q: What are some common mistakes to avoid when using the point-slope equation?

A: Some common mistakes to avoid when using the point-slope equation include:

• Not isolating the $y$ term on the left-hand side
• Not identifying the slope correctly
• Not finding a point on the line correctly
• Not using the correct values for $x_1$ and $y_1$

Q: How do I apply the point-slope equation to real-world problems?

A: The point-slope equation has many real-world applications, including:

• Finding the slope of a line given two points
• Finding the equation of a line given a point and the slope
• Solving linear equations
• Graphing lines

Q: What are some common applications of the point-slope equation?

A: Some common applications of the point-slope equation include:

• Physics: Finding the equation of motion of an object
• Engineering: Designing and building structures
• Economics: Modeling economic systems
• Computer Science: Writing algorithms and programs

Q: Can I use the point-slope equation to solve quadratic equations?

A: No, the point-slope equation is used to solve linear equations, not quadratic equations. Quadratic equations require a different set of techniques and formulas to solve.

Q: What are some resources for learning more about the point-slope equation?

A: Some resources for learning more about the point-slope equation include:

• Khan Academy: Point-Slope Form
• Mathway: Point-Slope Form
• Wolfram Alpha: Point-Slope Form
• Algebra textbooks and online resources

Conclusion

The point-slope equation is a powerful tool for solving linear equations and finding the slope and coordinates of a point on the line. By understanding the point-slope equation and its applications, you can solve a wide range of problems in mathematics and real-world situations.