What Is The Slope Of The Line Whose Equation Is $y - 4 = \frac{5}{2}(x - 2$\]?A. $\frac{5}{2}$

What is the Slope of the Line Whose Equation is $y - 4 = \frac{5}{2}(x - 2)$?

Understanding the Equation of a Line

The equation of a line in slope-intercept form is given by $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. However, the given equation is in a different form, which is $y - 4 = \frac{5}{2}(x - 2)$. To find the slope of this line, we need to rewrite the equation in slope-intercept form.

Rewriting the Equation in Slope-Intercept Form

To rewrite the equation in slope-intercept form, we need to isolate $y$ on one side of the equation. We can do this by adding $4$ to both sides of the equation:

$$y - 4 + 4 = \frac{5}{2}(x - 2) + 4$$

This simplifies to:

$$y = \frac{5}{2}(x - 2) + 4$$

Simplifying the Equation

To simplify the equation further, we can distribute the $\frac{5}{2}$ to the terms inside the parentheses:

$$y = \frac{5}{2}x - 5 + 4$$

This simplifies to:

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

Finding the Slope

Now that we have the equation in slope-intercept form, we can easily identify the slope. The slope is the coefficient of the $x$ term, which is $\frac{5}{2}$.

Conclusion

In conclusion, the slope of the line whose equation is $y - 4 = \frac{5}{2}(x - 2)$ is $\frac{5}{2}$. This is because the equation can be rewritten in slope-intercept form as $y = \frac{5}{2}x - 1$, where the slope is the coefficient of the $x$ term.

The Importance of Slope

The slope of a line is an important concept in mathematics, particularly in algebra and geometry. It represents the rate of change of the line, and it can be used to determine the direction and steepness of the line. In this case, the slope of $\frac{5}{2}$ indicates that the line is steep and has a positive rate of change.

Real-World Applications

The concept of slope has many real-world applications. For example, in physics, the slope of a line can be used to represent the velocity of an object. In economics, the slope of a line can be used to represent the rate of change of a quantity, such as the price of a commodity. In engineering, the slope of a line can be used to represent the angle of a slope or the steepness of a ramp.

Common Misconceptions

There are several common misconceptions about the slope of a line. One misconception is that the slope is always positive. However, the slope can be positive, negative, or zero, depending on the direction and steepness of the line. Another misconception is that the slope is always a fraction. However, the slope can be a fraction, a whole number, or a decimal, depending on the equation of the line.

Tips and Tricks

Here are some tips and tricks for finding the slope of a line:

• Always rewrite the equation in slope-intercept form before finding the slope.
• Identify the coefficient of the $x$ term as the slope.
• Use the slope to determine the direction and steepness of the line.
• Be careful when dealing with fractions and decimals, as they can affect the slope.

Conclusion

In conclusion, the slope of the line whose equation is $y - 4 = \frac{5}{2}(x - 2)$ is $\frac{5}{2}$. This is because the equation can be rewritten in slope-intercept form as $y = \frac{5}{2}x - 1$, where the slope is the coefficient of the $x$ term. The concept of slope has many real-world applications, and it is an important concept in mathematics. By following the tips and tricks outlined above, you can easily find the slope of a line and understand its significance.
What is the Slope of the Line Whose Equation is $y - 4 = \frac{5}{2}(x - 2)$?

Q&A: Frequently Asked Questions About Slope

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, you need to rewrite the equation of the line in slope-intercept form, which is $y = mx + b$. The slope is then the coefficient of the $x$ term, which is $m$.

Q: What is the difference between slope and rate of change?

A: Slope and rate of change are related but distinct concepts. The slope of a line represents the rate of change of the line, but it is a more specific measure that takes into account the direction and steepness of the line.

Q: Can the slope of a line be negative?

A: Yes, the slope of a line can be negative. A negative slope indicates that the line is decreasing as $x$ increases.

Q: Can the slope of a line be zero?

A: Yes, the slope of a line can be zero. A zero slope indicates that the line is horizontal and does not change as $x$ increases.

Q: How do I determine the direction of a line from its slope?

A: To determine the direction of a line from its slope, you can use the following rules:

• If the slope is positive, the line is increasing as $x$ increases.
• If the slope is negative, the line is decreasing as $x$ increases.
• If the slope is zero, the line is horizontal.

Q: How do I determine the steepness of a line from its slope?

A: To determine the steepness of a line from its slope, you can use the following rules:

• If the slope is large (e.g. 2 or 3), the line is steep.
• If the slope is small (e.g. 0.5 or 0.2), the line is not steep.

Q: Can the slope of a line be a fraction?

A: Yes, the slope of a line can be a fraction. For example, the slope of the line $y = \frac{1}{2}x + 3$ is $\frac{1}{2}$.

Q: Can the slope of a line be a decimal?

A: Yes, the slope of a line can be a decimal. For example, the slope of the line $y = 0.5x + 3$ is $0.5$.

Q: How do I use the slope of a line in real-world applications?

A: The slope of a line can be used in a variety of real-world applications, such as:

• Calculating the velocity of an object
• Determining the rate of change of a quantity
• Modeling the behavior of a system

Q: What are some common mistakes to avoid when finding the slope of a line?

A: Some common mistakes to avoid when finding the slope of a line include:

• Not rewriting the equation in slope-intercept form
• Not identifying the coefficient of the $x$ term as the slope
• Not considering the direction and steepness of the line

Conclusion

In conclusion, the slope of a line is an important concept in mathematics that has many real-world applications. By understanding how to find the slope of a line and how to use it in different contexts, you can gain a deeper understanding of the world around you.