Write An Equation In Standard Form That Models The Statement:The Difference Between Fourteen $x$ And Twenty-four $y$ Is Forty-four.A. $14x - 24y = 44$ B. $(14 + X) - (24 + Y) = 44$ C. $24y = -14x - 44$

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Understanding the Problem

In mathematics, equations are used to represent real-world situations and relationships between variables. Standard form equations are a way to express these relationships in a concise and organized manner. In this article, we will explore how to write an equation in standard form that models a given statement.

The Statement: Difference between Two Variables

The statement we will be working with is: "The difference between fourteen $x$ and twenty-four $y$ is forty-four." This statement can be interpreted as an equation, where the difference between $14x$ and $24y$ is equal to $44$. Our goal is to write this equation in standard form.

Option A: The Correct Standard Form Equation

The correct standard form equation for the given statement is:

14x−24y=4414x - 24y = 44

This equation represents the difference between $14x$ and $24y$ as $44$. The variables $x$ and $y$ are represented by their coefficients, which are $14$ and $-24$, respectively. The constant term on the right-hand side of the equation is $44$.

Option B: Incorrect Standard Form Equation

The equation (14+x)−(24+y)=44(14 + x) - (24 + y) = 44 is not in standard form. This equation is an example of a verbal expression, where the variables are added to the constants. To write this equation in standard form, we need to distribute the negative sign to the terms inside the parentheses:

14+x−24−y=4414 + x - 24 - y = 44

Simplifying this equation, we get:

−10−y+x=44-10 - y + x = 44

This equation is still not in standard form, as the variables are not represented by their coefficients. To write this equation in standard form, we need to rearrange the terms:

−y+x=54-y + x = 54

Option C: Incorrect Standard Form Equation

The equation 24y=−14x−4424y = -14x - 44 is also not in standard form. This equation is an example of a verbal expression, where the variables are multiplied by their coefficients. To write this equation in standard form, we need to isolate the variables on one side of the equation:

24y+14x=−4424y + 14x = -44

This equation is still not in standard form, as the variables are not represented by their coefficients. To write this equation in standard form, we need to rearrange the terms:

14x−24y=4414x - 24y = 44

Conclusion

In conclusion, the correct standard form equation for the given statement is 14x−24y=4414x - 24y = 44. This equation represents the difference between $14x$ and $24y$ as $44$. The variables $x$ and $y$ are represented by their coefficients, which are $14$ and $-24$, respectively. The constant term on the right-hand side of the equation is $44$.

Tips for Writing Standard Form Equations

When writing standard form equations, remember to:

  • Represent the variables by their coefficients
  • Keep the variables on one side of the equation
  • Keep the constants on the other side of the equation
  • Use the correct signs for the coefficients and constants

By following these tips, you can write standard form equations that accurately represent real-world situations and relationships between variables.

Common Mistakes to Avoid

When writing standard form equations, be careful not to make the following mistakes:

  • Representing the variables by their values instead of their coefficients
  • Keeping the variables and constants on the same side of the equation
  • Using the wrong signs for the coefficients and constants

By avoiding these mistakes, you can write standard form equations that are accurate and concise.

Real-World Applications

Standard form equations have many real-world applications, including:

  • Modeling population growth and decline
  • Representing financial transactions and budgets
  • Describing physical systems and relationships between variables

By understanding how to write standard form equations, you can apply mathematical concepts to real-world situations and make informed decisions.

Practice Problems

To practice writing standard form equations, try the following problems:

  1. Write an equation in standard form that represents the statement: "The sum of $12x$ and $18y$ is $36$."
  2. Write an equation in standard form that represents the statement: "The difference between $20x$ and $15y$ is $-30$."
  3. Write an equation in standard form that represents the statement: "The product of $8x$ and $12y$ is $96$."

Q: What is a standard form equation?

A: A standard form equation is a mathematical equation that represents a relationship between variables in a concise and organized manner. It is typically written in the form of Ax + By = C, where A, B, and C are constants, and x and y are variables.

Q: How do I write a standard form equation?

A: To write a standard form equation, follow these steps:

  1. Identify the variables and their coefficients in the equation.
  2. Represent the variables by their coefficients.
  3. Keep the variables on one side of the equation.
  4. Keep the constants on the other side of the equation.
  5. Use the correct signs for the coefficients and constants.

Q: What are some common mistakes to avoid when writing standard form equations?

A: Some common mistakes to avoid when writing standard form equations include:

  • Representing the variables by their values instead of their coefficients.
  • Keeping the variables and constants on the same side of the equation.
  • Using the wrong signs for the coefficients and constants.

Q: How do I determine the coefficients of a standard form equation?

A: To determine the coefficients of a standard form equation, follow these steps:

  1. Identify the variables and their values in the equation.
  2. Represent the variables by their coefficients.
  3. Use the correct signs for the coefficients and constants.

Q: What are some real-world applications of standard form equations?

A: Standard form equations have many real-world applications, including:

  • Modeling population growth and decline.
  • Representing financial transactions and budgets.
  • Describing physical systems and relationships between variables.

Q: How do I solve a standard form equation?

A: To solve a standard form equation, follow these steps:

  1. Isolate the variables on one side of the equation.
  2. Use the correct signs for the coefficients and constants.
  3. Solve for the variables.

Q: What are some tips for writing and solving standard form equations?

A: Some tips for writing and solving standard form equations include:

  • Use the correct signs for the coefficients and constants.
  • Keep the variables and constants on opposite sides of the equation.
  • Use the distributive property to simplify the equation.
  • Check your work by plugging in the values of the variables.

Q: How do I graph a standard form equation?

A: To graph a standard form equation, follow these steps:

  1. Identify the coefficients of the equation.
  2. Determine the x and y intercepts of the equation.
  3. Plot the intercepts on a coordinate plane.
  4. Draw a line through the intercepts to represent the equation.

Q: What are some common mistakes to avoid when graphing standard form equations?

A: Some common mistakes to avoid when graphing standard form equations include:

  • Plotting the intercepts incorrectly.
  • Drawing the line through the intercepts incorrectly.
  • Not using the correct scale for the coordinate plane.

Q: How do I use standard form equations in real-world applications?

A: To use standard form equations in real-world applications, follow these steps:

  1. Identify the variables and their relationships in the problem.
  2. Write a standard form equation to represent the relationship.
  3. Solve the equation to find the values of the variables.
  4. Use the values of the variables to make informed decisions.

Q: What are some examples of standard form equations in real-world applications?

A: Some examples of standard form equations in real-world applications include:

  • Modeling population growth and decline.
  • Representing financial transactions and budgets.
  • Describing physical systems and relationships between variables.

By following these steps and avoiding common mistakes, you can use standard form equations to solve real-world problems and make informed decisions.