Select The Correct Answer.Niall Owes $\$ 187$ To His Cousin. He Hopes To Earn Some Money By Painting A House. He Gets Paid $\$ 34$[/tex\] For Every 2 Hours He Paints. Which Equation Of A Line Models $y$, The Amount
Introduction
In this problem, we are given that Niall owes $187 to his cousin and hopes to earn some money by painting a house. He gets paid $34 for every 2 hours he paints. We need to find the equation of a line that models y, the amount Niall earns, in terms of the number of hours he paints. This problem involves understanding the concept of linear equations and how they can be used to model real-world situations.
Understanding the Concept of Linear Equations
A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line represents the rate of change of the variable(s) with respect to the independent variable, while the y-intercept represents the value of the dependent variable when the independent variable is equal to zero.
Identifying the Independent and Dependent Variables
In this problem, the independent variable is the number of hours Niall paints, and the dependent variable is the amount he earns. We can represent the number of hours Niall paints as x and the amount he earns as y.
Determining the Slope of the Line
We are given that Niall gets paid $34 for every 2 hours he paints. This means that for every 2 hours he paints, his earnings increase by $34. To find the slope of the line, we need to determine the rate of change of his earnings with respect to the number of hours he paints.
Since Niall gets paid $34 for every 2 hours he paints, we can say that his earnings increase by $17 for every 1 hour he paints (since $34 Ă· 2 = $17). Therefore, the slope of the line is 17.
Determining the Y-Intercept of the Line
We are given that Niall owes $187 to his cousin. This means that when Niall paints for 0 hours, his earnings are equal to $187 (since he owes $187 to his cousin). Therefore, the y-intercept of the line is 187.
Writing the Equation of the Line
Now that we have determined the slope and y-intercept of the line, we can write the equation of the 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 case, the equation of the line is:
y = 17x + 187
Conclusion
In this problem, we were given that Niall owes $187 to his cousin and hopes to earn some money by painting a house. He gets paid $34 for every 2 hours he paints. We needed to find the equation of a line that models y, the amount Niall earns, in terms of the number of hours he paints. We determined the slope and y-intercept of the line and wrote the equation of the line as y = 17x + 187.
Key Takeaways
- A linear equation is an equation in which the highest power of the variable(s) is 1.
- The slope of a line represents the rate of change of the variable(s) with respect to the independent variable.
- The y-intercept of a line represents the value of the dependent variable when the independent variable is equal to zero.
- 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.
Real-World Applications
Linear equations have many real-world applications, including:
- Modeling the cost of goods and services
- Determining the rate of change of a quantity with respect to time
- Finding the equation of a line that passes through a given point and has a given slope
- Solving systems of linear equations
Common Mistakes to Avoid
When working with linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:
- Not checking the units of the variables
- Not checking the signs of the coefficients
- Not checking the equation for extraneous solutions
- Not using the correct method to solve the equation
Conclusion
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: What is the slope of a line?
A: The slope of a line represents the rate of change of the variable(s) with respect to the independent variable. It is a measure of how much the dependent variable changes when the independent variable changes by one unit.
Q: What is the y-intercept of a line?
A: The y-intercept of a line represents the value of the dependent variable when the independent variable is equal to zero. It is the point at which the line intersects the y-axis.
Q: How do I write the equation of a line?
A: To write the equation of a line, you need to determine the slope and y-intercept of the 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.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. In other words, a linear equation can be written in the form y = mx + b, while a quadratic equation can be written in the form y = ax^2 + bx + c.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Q: What is the difference between a linear equation and a system of linear equations?
A: A linear equation is a single equation that can be written in the form y = mx + b, while a system of linear equations is a set of two or more linear equations that are solved simultaneously.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to find the values of the variables that satisfy all of the equations in the system. You can do this by using substitution or elimination methods.
Q: What are some real-world applications of linear equations?
A: Linear equations have many real-world applications, including:
- Modeling the cost of goods and services
- Determining the rate of change of a quantity with respect to time
- Finding the equation of a line that passes through a given point and has a given slope
- Solving systems of linear equations
Q: What are some common mistakes to avoid when working with linear equations?
A: Some common mistakes to avoid when working with linear equations include:
- Not checking the units of the variables
- Not checking the signs of the coefficients
- Not checking the equation for extraneous solutions
- Not using the correct method to solve the equation
Conclusion
In conclusion, linear equations are an important concept in mathematics that have many real-world applications. By understanding the concept of linear equations and how to write the equation of a line, we can model real-world situations and solve problems that involve linear relationships.