For A Given Input Value $x$, The Function $f$ Outputs A Value $y$ To Satisfy The Following Equation:$-5x - 4y = -8$Write A Formula For $f(x$\] In Terms Of $x$.$f(x) =$ $\square$

Introduction

In mathematics, solving linear equations is a fundamental concept that helps us understand the relationship between variables. Given an equation, we can use algebraic techniques to isolate the variable and find its value. In this article, we will focus on solving the linear equation $-5x - 4y = -8$ to find the formula for $f(x)$.

Understanding the Equation

The given equation is a linear equation in two variables, $x$ and $y$. The equation is in the form $ax + by = c$, where $a$, $b$, and $c$ are constants. In this case, $a = -5$, $b = -4$, and $c = -8$. Our goal is to find the value of $y$ in terms of $x$.

Solving for y

To solve for $y$, we can use the method of substitution or elimination. In this case, we will use the substitution method. We can start by isolating $y$ on one side of the equation.

Step 1: Add 5x to both sides of the equation

Adding $5x$ to both sides of the equation gives us:

$$-4y = -8 + 5x$$

Step 2: Divide both sides of the equation by -4

Dividing both sides of the equation by $-4$ gives us:

$$y = \frac{-8 + 5x}{-4}$$

Step 3: Simplify the expression

Simplifying the expression gives us:

$$y = \frac{5x - 8}{4}$$

Finding the Formula for f(x)

Now that we have found the value of $y$ in terms of $x$, we can write the formula for $f(x)$.

$$f(x) = \frac{5x - 8}{4}$$

This formula represents the output of the function $f$ for a given input value $x$.

Conclusion

In this article, we solved the linear equation $-5x - 4y = -8$ to find the formula for $f(x)$. We used the substitution method to isolate $y$ on one side of the equation and then simplified the expression to find the final formula. The formula for $f(x)$ is $\frac{5x - 8}{4}$.

Example Use Case

Suppose we want to find the value of $f(2)$. We can plug in $x = 2$ into the formula for $f(x)$:

$$f(2) = \frac{5(2) - 8}{4}$$

$$f(2) = \frac{10 - 8}{4}$$

$$f(2) = \frac{2}{4}$$

$$f(2) = \frac{1}{2}$$

Therefore, the value of $f(2)$ is $\frac{1}{2}$.

Applications of Linear Equations

Linear equations have numerous applications in various fields, including physics, engineering, economics, and computer science. Some examples of applications include:

• Physics: Linear equations are used to describe the motion of objects under the influence of forces.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about economic outcomes.
• Computer Science: Linear equations are used in algorithms and data structures, such as linear programming and graph theory.

Final Thoughts

In conclusion, solving linear equations is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to solve linear equations, we can gain insights into the relationships between variables and make predictions about real-world phenomena. The formula for $f(x)$, which we derived in this article, is a simple example of a linear equation that can be used to model a wide range of phenomena.

References

• [1] "Linear Equations" by Khan Academy
• [2] "Solving Linear Equations" by Mathway
• [3] "Linear Programming" by MIT OpenCourseWare

Further Reading

• "Linear Algebra" by Gilbert Strang
• "Introduction to Linear Algebra" by Gilbert Strang
• "Linear Equations and Inequalities" by Michael Artin
  

Introduction

In our previous article, we solved the linear equation $-5x - 4y = -8$ to find the formula for $f(x)$. In this article, we will answer some common questions related to solving linear equations and finding the formula for $f(x)$.

Q&A

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 $ax + by = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the method of substitution or elimination. The method of substitution involves isolating one variable on one side of the equation and then substituting that expression into the other side of the equation. The method of elimination involves adding or subtracting the same value to both sides of the equation to eliminate one variable.

Q: What is the formula for f(x)?

A: The formula for $f(x)$ is $\frac{5x - 8}{4}$. This formula represents the output of the function $f$ for a given input value $x$.

Q: How do I find the value of f(x) for a given input value x?

A: To find the value of $f(x)$ for a given input value $x$, you can plug in the value of $x$ into the formula for $f(x)$. For example, if you want to find the value of $f(2)$, you can plug in $x = 2$ into the formula:

$$f(2) = \frac{5(2) - 8}{4}$$

$$f(2) = \frac{10 - 8}{4}$$

$$f(2) = \frac{2}{4}$$

$$f(2) = \frac{1}{2}$$

Q: What are some common applications of linear equations?

A: Linear equations have numerous applications in various fields, including physics, engineering, economics, and computer science. Some examples of applications include:

• Physics: Linear equations are used to describe the motion of objects under the influence of forces.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about economic outcomes.
• Computer Science: Linear equations are used in algorithms and data structures, such as linear programming and graph theory.

Q: How do I know if a linear equation has a solution?

A: A linear equation has a solution if the coefficients of the variables are not equal to zero. In other words, if the equation is in the form $ax + by = c$, where $a$, $b$, and $c$ are constants, then the equation has a solution if $a \neq 0$ and $b \neq 0$.

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. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation, while the equation $2x + 3y = 5$ is a linear equation.

Conclusion

In this article, we answered some common questions related to solving linear equations and finding the formula for $f(x)$. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.

Further Reading

• "Linear Equations" by Khan Academy
• "Solving Linear Equations" by Mathway
• "Linear Programming" by MIT OpenCourseWare

References

• [1] "Linear Equations" by Khan Academy
• [2] "Solving Linear Equations" by Mathway
• [3] "Linear Programming" by MIT OpenCourseWare

Final Thoughts

In conclusion, solving linear equations is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to solve linear equations, we can gain insights into the relationships between variables and make predictions about real-world phenomena. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.