For The Equation $6x + 5y = 30$, Complete The Following Tasks:(a) Write It In Slope-intercept Form. - The Slope-intercept Form Of The Equation Is $y = -\frac{6}{5}x + 6$. (Simplify Your Answer. Use Integers Or Fractions
Introduction
Linear equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. One of the most common forms of linear equations is the slope-intercept form, which is represented as y = mx + b, where m is the slope and b is the y-intercept. In this article, we will focus on converting the given linear equation $6x + 5y = 30$ to its slope-intercept form.
Understanding the Slope-Intercept Form
The slope-intercept form of a linear equation is a powerful tool for analyzing and solving linear equations. It provides a clear and concise representation of the equation, making it easier to understand and work with. The slope-intercept form is characterized by the presence of the slope (m) and the y-intercept (b), which are the coefficients of the x and y terms, respectively.
Converting the Given Equation to Slope-Intercept Form
To convert the given equation $6x + 5y = 30$ to its slope-intercept form, we need to isolate the y term on one side of the equation. We can do this by subtracting 6x from both sides of the equation, which gives us:
Next, we need to divide both sides of the equation by 5 to isolate the y term. This gives us:
Simplifying the Equation
The equation $y = -\frac{6}{5}x + 6$ is already in its simplest form, and it represents the slope-intercept form of the given linear equation. The slope of the equation is -6/5, and the y-intercept is 6.
Interpreting the Results
The slope-intercept form of the equation provides valuable information about the behavior of the linear equation. The slope (m) indicates the rate of change of the equation, while the y-intercept (b) indicates the point at which the equation intersects the y-axis. In this case, the slope is -6/5, which means that the equation decreases at a rate of 6/5 units for every unit increase in x. The y-intercept is 6, which means that the equation intersects the y-axis at the point (0, 6).
Conclusion
In conclusion, converting the given linear equation $6x + 5y = 30$ to its slope-intercept form requires isolating the y term on one side of the equation and then dividing both sides by the coefficient of the y term. The resulting equation $y = -\frac{6}{5}x + 6$ represents the slope-intercept form of the given linear equation, and it provides valuable information about the behavior of the equation.
Applications of Slope-Intercept Form
The slope-intercept form of a linear equation has numerous applications in various fields, including physics, engineering, and economics. Some of the key applications of slope-intercept form include:
- Linear Regression: The slope-intercept form is used to model the relationship between two variables in linear regression analysis.
- Physics: The slope-intercept form is used to describe the motion of objects under the influence of gravity or other forces.
- Engineering: The slope-intercept form is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: The slope-intercept form is used to model the relationship between economic variables, such as supply and demand.
Tips and Tricks
Here are some tips and tricks for working with slope-intercept form:
- Use the slope-intercept form to identify the slope and y-intercept of a linear equation.
- Use the slope-intercept form to graph a linear equation.
- Use the slope-intercept form to solve linear equations.
- Use the slope-intercept form to model real-world problems.
Common Mistakes
Here are some common mistakes to avoid when working with slope-intercept form:
- Not isolating the y term on one side of the equation.
- Not dividing both sides of the equation by the coefficient of the y term.
- Not simplifying the equation.
- Not interpreting the results correctly.
Conclusion
In conclusion, the slope-intercept form of a linear equation is a powerful tool for analyzing and solving linear equations. By converting the given equation $6x + 5y = 30$ to its slope-intercept form, we can gain valuable insights into the behavior of the equation and its applications in various fields.
Introduction
In our previous article, we discussed how to convert a linear equation to its slope-intercept form. In this article, we will answer some frequently asked questions about solving linear equations and provide additional tips and tricks for working with slope-intercept form.
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 and b is the y-intercept.
Q: How do I convert a linear equation to its slope-intercept form?
A: To convert a linear equation to its slope-intercept form, you need to isolate the y term on one side of the equation and then divide both sides by the coefficient of the y term.
Q: What is the slope of a linear equation?
A: The slope of a linear equation is the coefficient of the x term, which represents the rate of change of the equation.
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the point at which the equation intersects the y-axis.
Q: How do I graph a linear equation in slope-intercept form?
A: To graph a linear equation in slope-intercept form, you can use the slope and y-intercept to plot two points on the graph and then draw a line through them.
Q: How do I solve a linear equation in slope-intercept form?
A: To solve a linear equation in slope-intercept form, you can set the equation equal to a specific value and then solve for x.
Q: What are some common mistakes to avoid when working with slope-intercept form?
A: Some common mistakes to avoid when working with slope-intercept form include not isolating the y term on one side of the equation, not dividing both sides of the equation by the coefficient of the y term, not simplifying the equation, and not interpreting the results correctly.
Tips and Tricks
Here are some additional tips and tricks for working with slope-intercept form:
- Use the slope-intercept form to identify the slope and y-intercept of a linear equation.
- Use the slope-intercept form to graph a linear equation.
- Use the slope-intercept form to solve linear equations.
- Use the slope-intercept form to model real-world problems.
- Make sure to simplify the equation before interpreting the results.
- Use the slope-intercept form to identify the equation of a line that passes through two points.
Common Applications
Here are some common applications of slope-intercept form:
- Linear Regression: The slope-intercept form is used to model the relationship between two variables in linear regression analysis.
- Physics: The slope-intercept form is used to describe the motion of objects under the influence of gravity or other forces.
- Engineering: The slope-intercept form is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: The slope-intercept form is used to model the relationship between economic variables, such as supply and demand.
Conclusion
In conclusion, the slope-intercept form of a linear equation is a powerful tool for analyzing and solving linear equations. By understanding the slope-intercept form and its applications, you can gain valuable insights into the behavior of linear equations and their uses in various fields.
Additional Resources
Here are some additional resources for learning more about slope-intercept form:
- Textbooks: There are many textbooks available that cover the topic of slope-intercept form in detail.
- Online Resources: There are many online resources available that provide tutorials and examples of slope-intercept form.
- Practice Problems: There are many practice problems available that can help you practice working with slope-intercept form.
Conclusion
In conclusion, the slope-intercept form of a linear equation is a fundamental concept in mathematics that has numerous applications in various fields. By understanding the slope-intercept form and its applications, you can gain valuable insights into the behavior of linear equations and their uses in various fields.