Which Of The Following Represents $3x - 5y + 10 = 0$ Written In Slope-intercept Form?A. $y = -\frac{3}{5}x + 2$ B. $ Y = 3 5 X − 2 Y = \frac{3}{5}x - 2 Y = 5 3 X − 2 [/tex] C. $y = \frac{3}{5}x + 2$
Introduction
In mathematics, linear equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. One of the essential skills in solving linear equations is converting them into slope-intercept form, which is a powerful tool for analyzing and understanding the behavior of linear functions. In this article, we will explore how to convert the given linear equation $3x - 5y + 10 = 0$ into slope-intercept form and identify the correct representation among the given options.
Understanding Slope-Intercept Form
Slope-intercept form is a way of writing a linear equation in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. This form is particularly useful for analyzing the behavior of linear functions, as it allows us to easily identify the slope and y-intercept of the line.
Converting the Given Equation to Slope-Intercept Form
To convert the given equation $3x - 5y + 10 = 0$ into slope-intercept form, we need to isolate the variable $y$. We can do this by subtracting $3x$ from both sides of the equation and then dividing both sides by $-5$.
Subtract $3x$ from both sides:
Subtract $10$ from both sides:
Divide both sides by $-5$:
Analyzing the Options
Now that we have converted the given equation to slope-intercept form, we can analyze the options to identify the correct representation.
- Option A: $y = -\frac{3}{5}x + 2$
- Option B: $y = \frac{3}{5}x - 2$
- Option C: $y = \frac{3}{5}x + 2$
Conclusion
Based on our analysis, we can see that the correct representation of the given equation $3x - 5y + 10 = 0$ in slope-intercept form is:
This is the only option that matches the slope-intercept form of the equation, with a slope of $\frac{3}{5}$ and a y-intercept of $2$.
Final Answer
The final answer is:
Additional Tips and Resources
- To convert a linear equation to slope-intercept form, isolate the variable $y$ by subtracting or adding terms to both sides of the equation.
- Use the slope-intercept form to analyze the behavior of linear functions, including the slope and y-intercept.
- Practice converting linear equations to slope-intercept form to develop your skills and build your confidence.
References
Related Topics
- Graphing Linear Equations
- Solving Systems of Linear Equations
- Linear Functions
Solving Linear Equations: Q&A =============================
Introduction
In our previous article, we explored how to convert the given linear equation $3x - 5y + 10 = 0$ into slope-intercept form and identified the correct representation among the given options. In this article, we will provide a Q&A section to help you better understand the concepts and provide additional insights.
Q&A
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.
Q: How do I convert a linear equation to slope-intercept form?
A: To convert a linear equation to slope-intercept form, isolate the variable $y$ by subtracting or adding terms to both sides of the equation.
Q: What is the slope of the line in the equation $y = \frac{3}{5}x + 2$?
A: The slope of the line in the equation $y = \frac{3}{5}x + 2$ is $\frac{3}{5}$.
Q: What is the y-intercept of the line in the equation $y = \frac{3}{5}x + 2$?
A: The y-intercept of the line in the equation $y = \frac{3}{5}x + 2$ is $2$.
Q: How do I determine the slope and y-intercept of a linear equation?
A: To determine the slope and y-intercept of a linear equation, convert the equation to slope-intercept form and identify the values of $m$ and $b$.
Q: What is the difference between the slope and y-intercept?
A: The slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.
Q: Can I use the slope-intercept form to graph a linear equation?
A: Yes, you can use the slope-intercept form to graph a linear equation by plotting the y-intercept and using the slope to determine the direction and steepness of the line.
Q: How do I use the slope-intercept form to analyze the behavior of a linear function?
A: You can use the slope-intercept form to analyze the behavior of a linear function by examining the values of $m$ and $b$ and determining how they affect the graph of the function.
Additional Tips and Resources
- Practice converting linear equations to slope-intercept form to develop your skills and build your confidence.
- Use the slope-intercept form to analyze the behavior of linear functions, including the slope and y-intercept.
- Explore the relationship between the slope and y-intercept and how they affect the graph of a linear function.
References
Related Topics
Conclusion
In this Q&A article, we provided additional insights and answers to common questions about solving linear equations and converting them to slope-intercept form. We hope this article has been helpful in your understanding of the concepts and has provided you with the tools and resources you need to succeed in mathematics.