Solve For R R R : − 132 = 7 ( 5 R − 8 ) + 3 R -132 = 7(5r - 8) + 3r − 132 = 7 ( 5 R − 8 ) + 3 R

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable $r$. We will use the given equation $-132 = 7(5r - 8) + 3r$ as an example to demonstrate the step-by-step process of solving for $r$.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand what it represents. The equation is in the form of a linear equation, where the variable $r$ is isolated on one side of the equation. The equation is:

$$-132 = 7(5r - 8) + 3r$$

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient $7$ to the terms inside the parentheses. This will help us simplify the equation and make it easier to work with.

$$-132 = 7(5r) - 7(8) + 3r$$

Step 2: Simplify the Equation

Now that we have distributed the coefficient, we can simplify the equation by combining like terms.

$$-132 = 35r - 56 + 3r$$

Step 3: Combine Like Terms

The next step is to combine the like terms on the right-hand side of the equation. In this case, we have two terms with the variable $r$, which we can combine by adding their coefficients.

$$-132 = (35r + 3r) - 56$$

Step 4: Simplify the Coefficient

Now that we have combined the like terms, we can simplify the coefficient of the variable $r$ by adding its values.

$$-132 = 38r - 56$$

Step 5: Add 56 to Both Sides

The next step is to add $56$ to both sides of the equation to isolate the term with the variable $r$.

$$-132 + 56 = 38r - 56 + 56$$

Step 6: Simplify the Equation

Now that we have added $56$ to both sides, we can simplify the equation by combining like terms.

$$-76 = 38r$$

Step 7: Divide Both Sides

The final step is to divide both sides of the equation by the coefficient of the variable $r$ to solve for $r$.

$$\frac{-76}{38} = \frac{38r}{38}$$

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation to isolate the variable $r$. We have used the given equation $-132 = 7(5r - 8) + 3r$ as an example to illustrate the process. By following these steps, we can solve for $r$ and find the value of the variable.

Final Answer

The final answer to the equation $-132 = 7(5r - 8) + 3r$ is:

$$r = -2$$

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

• Always start by simplifying the equation and combining like terms.
• Use the distributive property to distribute coefficients to terms inside parentheses.
• Add or subtract the same value to both sides of the equation to isolate the term with the variable.
• Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation to isolate the variable $r$. However, we understand that sometimes, it's not just about following a set of steps, but also about understanding the underlying concepts and principles. In this article, we will address some of the most frequently asked questions about solving linear equations, and provide answers to help you better understand the subject.

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 of $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I know if an equation is linear or not?

A: To determine if an equation is linear or not, you need to look at the highest power of the variable(s). If the highest power is 1, then the equation is linear. If the highest power is greater than 1, then the equation is not linear.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows you to distribute a coefficient to the terms inside parentheses. In other words, it allows you to multiply a coefficient by each term inside the parentheses.

Q: How do I use the distributive property to solve a linear equation?

A: To use the distributive property to solve a linear equation, you need to multiply the coefficient by each term inside the parentheses. This will help you simplify the equation and make it easier to work with.

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 of $ax + b = c$, while a quadratic equation can be written in the form of $ax^2 + bx + c = 0$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula, which is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that allows you to solve a quadratic equation. It is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of $a$, $b$, and $c$ into the formula. This will give you the solutions to the equation.

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

A: A linear equation is an equation in which the highest power of the variable(s) is 1, 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 use methods such as substitution or elimination to find the values of the variables.

Conclusion

In this article, we have addressed some of the most frequently asked questions about solving linear equations, and provided answers to help you better understand the subject. We hope that this article has been helpful in clarifying any doubts you may have had about solving linear equations.

Final Tips and Tricks

Here are some final tips and tricks to help you solve linear equations:

• Always start by simplifying the equation and combining like terms.
• Use the distributive property to distribute coefficients to terms inside parentheses.
• Add or subtract the same value to both sides of the equation to isolate the term with the variable.
• Divide both sides of the equation by the coefficient of the variable to solve for the variable.
• Use the quadratic formula to solve quadratic equations.

By following these tips and tricks, you can become proficient in solving linear equations and master the art of algebra.