What Is The Slope Of The Line Described By The Equation Below? Y − 5 = − 3 ( X − 17 Y - 5 = -3(x - 17 Y − 5 = − 3 ( X − 17 ]A. 3 B. 5 C. -3 D. -5
What is the Slope of a Line Described by the Equation?
The slope of a line is a fundamental concept in mathematics that describes the rate of change of a variable with respect to another variable. In this article, we will explore the concept of slope and how to find it using a given equation.
What is the Slope of a Line?
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. The slope is often denoted by the letter 'm' and is a key concept in graphing and analyzing linear equations.
The Equation of a Line
The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. In this equation, m represents the rate of change of the line, and b represents the point where the line intersects the y-axis.
Slope-Intercept Form
The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. This form is useful for graphing and analyzing linear equations.
Finding the Slope of a Line
To find the slope of a line, we can use the slope-intercept form of the equation. The slope is the coefficient of the x-term, which is the number that multiplies the x-variable.
Example: Finding the Slope of a Line
Let's consider the equation y - 5 = -3(x - 17). To find the slope of this line, we need to rewrite the equation in slope-intercept form.
Step 1: Rewrite the Equation
y - 5 = -3(x - 17)
Step 2: Distribute the -3
y - 5 = -3x + 51
Step 3: Add 5 to Both Sides
y = -3x + 56
Step 4: Identify the Slope
The slope of the line is the coefficient of the x-term, which is -3.
Conclusion
In this article, we have explored the concept of slope and how to find it using a given equation. We have seen that the slope of a line is a measure of how steep it is and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. We have also seen how to rewrite an equation in slope-intercept form and identify the slope.
Answer
The slope of the line described by the equation y - 5 = -3(x - 17) is C. -3.
Key Takeaways
- The slope of a line is a measure of how steep it is.
- The slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
- The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
- To find the slope of a line, we can use the slope-intercept form of the equation and identify the coefficient of the x-term.
Frequently Asked Questions
- What is the slope of a line?
- How do I find the slope of a line?
- What is the slope-intercept form of a line?
- How do I rewrite an equation in slope-intercept form?
Related Topics
- Graphing linear equations
- Analyzing linear equations
- Slope-intercept form
- Linear equations
Conclusion
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 can use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept. You can also use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. This form is useful for graphing and analyzing linear equations.
Q: How do I rewrite an equation in slope-intercept form?
A: To rewrite an equation in slope-intercept form, you need to isolate the y-variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.
Q: What is the difference between the slope and the y-intercept?
A: The slope (m) is the coefficient of the x-term in the equation, while the y-intercept (b) is the value of the y-variable when the x-variable is equal to zero.
Q: How do I graph a line using its slope and y-intercept?
A: To graph a line using its slope and y-intercept, you can use the slope-intercept form of the equation, which is y = mx + b. You can plot the y-intercept on the y-axis and then use the slope to find the next point on the line.
Q: What is the significance of the slope in real-world applications?
A: The slope is an important concept in many real-world applications, such as finance, economics, and physics. It is used to describe the rate of change of a variable with respect to another variable, and is essential for making predictions and analyzing data.
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 the x-variable 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 the x-variable increases.
Q: Can the slope of a line be undefined?
A: Yes, the slope of a line can be undefined. An undefined slope indicates that the line is vertical and does not change as the y-variable increases.
Q: How do I determine the slope of a line from a graph?
A: To determine the slope of a line from a graph, you can use the following steps:
- Identify two points on the line.
- Calculate the vertical change (rise) between the two points.
- Calculate the horizontal change (run) between the two points.
- Divide the vertical change by the horizontal change to find the slope.
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 isolating the y-variable on one side of the equation.
- Not using the correct formula for the slope.
- Not checking for any errors in the calculation.
- Not considering the possibility of a negative or zero slope.
Q: How do I use the slope to make predictions about a line?
A: To use the slope to make predictions about a line, you can use the following steps:
- Identify the slope of the line.
- Use the slope to find the rate of change of the line.
- Use the rate of change to make predictions about the line.
Q: What are some real-world applications of the slope?
A: Some real-world applications of the slope include:
- Finance: The slope is used to describe the rate of return on an investment.
- Economics: The slope is used to describe the rate of change of a variable with respect to another variable.
- Physics: The slope is used to describe the rate of change of a variable with respect to another variable.
Conclusion
In conclusion, the slope of a line is an important concept in mathematics that describes the rate of change of a variable with respect to another variable. We have seen how to find the slope of a line using a given equation and have explored the concept of slope-intercept form. We hope that this article has provided a clear understanding of the concept of slope and how to find it.