Solve \[$16 + 31x = -13\$\].Write \[$x\$\] As A Fraction In Its Simplest Form.

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Introduction

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a given equation. In this article, we will focus on solving the equation ${16 + 31x = -13\$} and express the value of {x$}$ as a fraction in its simplest form.

Understanding the Equation

The given equation is a linear equation in one variable, {x$}$. The equation is in the form of {ax + b = c$}$, where {a$}$ is the coefficient of the variable, {b$}$ is the constant term, and {c$}$ is the constant term on the right-hand side. In this case, {a = 31$}$, {b = 16$}$, and {c = -13$}$.

Isolating the Variable

To solve the equation, we need to isolate the variable {x$}$ on one side of the equation. We can do this by subtracting the constant term {b$}$ from both sides of the equation. This will give us:

${16 + 31x - 16 = -13 - 16\$}

Simplifying the equation, we get:

${31x = -29\$}

Solving for {x$}$

Now that we have isolated the variable, we can solve for {x$}$ by dividing both sides of the equation by the coefficient of the variable, which is ${31\$}. This will give us:

{x = \frac{-29}{31}$}$

Simplifying the Fraction

The fraction {\frac{-29}{31}$}$ is already in its simplest form, as the numerator and denominator have no common factors. However, we can express the fraction in a more simplified form by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of {-29$}$ and ${31\$} is ${1\$}, so the fraction remains the same.

Conclusion

In conclusion, we have solved the equation ${16 + 31x = -13\$} and expressed the value of {x$}$ as a fraction in its simplest form. The solution is {x = \frac{-29}{31}$}$, which is a simplified fraction.

Frequently Asked Questions

  • What is the value of {x$}$ in the equation ${16 + 31x = -13\$}?
  • How do you solve a linear equation in one variable?
  • What is the simplest form of the fraction {\frac{-29}{31}$}$?

Final Answer

The final answer is {x = \frac{-29}{31}$}$.

Introduction

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a given equation. In this article, we will provide a Q&A section to help you better understand how to solve linear equations and address some common questions related to this topic.

Q1: What is a linear equation?

A linear equation is an equation in which the highest power of the variable is 1. It is typically written in the form of {ax + b = c$}$, where {a$}$ is the coefficient of the variable, {b$}$ is the constant term, and {c$}$ is the constant term on the right-hand side.

Q2: How do you solve a linear equation?

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. The goal is to get the variable by itself on one side of the equation.

Q3: What is the difference between a linear equation and a quadratic equation?

A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, ${2x + 3 = 5\$} is a linear equation, while {x^2 + 4x + 4 = 0$}$ is a quadratic equation.

Q4: How do you simplify a fraction in its simplest form?

To simplify a fraction in its simplest form, you need to find the greatest common divisor (GCD) of the numerator and denominator. You can then divide both the numerator and denominator by the GCD to get the simplified fraction.

Q5: What is the value of {x$}$ in the equation ${16 + 31x = -13\$}?

The value of {x$}$ in the equation ${16 + 31x = -13\$} is {x = \frac{-29}{31}$}$.

Q6: How do you solve an equation with a negative coefficient?

To solve an equation with a negative coefficient, you need to multiply both sides of the equation by -1 to get rid of the negative sign. For example, if you have the equation {-3x + 2 = 5$}$, you can multiply both sides by -1 to get ${3x - 2 = -5\$}.

Q7: What is the difference between a linear equation and a system of linear equations?

A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more linear equations with the same variable. For example, ${2x + 3 = 5\$} is a linear equation, while ${2x + 3 = 5\$} and {x - 2 = 3$}$ is a system of linear equations.

Q8: How do you solve a system of linear equations?

To solve a system of linear equations, you need to find the value of the variable that satisfies both equations. You can do this by using substitution or elimination methods.

Q9: What is the importance of solving linear equations?

Solving linear equations is an essential skill in mathematics and is used in a wide range of applications, including science, engineering, economics, and finance.

Q10: How can I practice solving linear equations?

You can practice solving linear equations by working on exercises and problems in your textbook or online resources. You can also try solving real-world problems that involve linear equations.

Final Answer

The final answer is that solving linear equations is an essential skill in mathematics that has a wide range of applications. By understanding how to isolate the variable in a given equation, you can solve linear equations and address common questions related to this topic.