Solve The Equation:${ -3(1 + 6r) = 14 - R }$

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Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a specific linear equation, $-3(1 + 6r) = 14 - r$, using step-by-step instructions.

Understanding the Equation

The given equation is $-3(1 + 6r) = 14 - r$. To solve this equation, we need to isolate the variable $r$. The equation involves a negative coefficient, which can be challenging to handle. However, with a clear understanding of the equation and a systematic approach, we can solve it easily.

Step 1: Distribute the Negative Coefficient

The first step in solving the equation is to distribute the negative coefficient $-3$ to the terms inside the parentheses. This will help us simplify the equation and make it easier to work with.

$$-3(1 + 6r) = -3(1) - 3(6r)$$

Step 2: Simplify the Equation

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

$$-3(1 + 6r) = -3 - 18r$$

Step 3: Move All Terms to One Side

To isolate the variable $r$, we need to move all the terms to one side of the equation. We can do this by adding $18r$ to both sides of the equation.

$$-3 - 18r = 14 - r$$

Step 4: Add $18r$ to Both Sides

By adding $18r$ to both sides of the equation, we have moved all the terms involving $r$ to the left-hand side.

$$-3 = 14 - r + 18r$$

Step 5: Combine Like Terms

Now that we have moved all the terms involving $r$ to the left-hand side, we can combine like terms.

$$-3 = 14 + 17r$$

Step 6: Subtract 14 from Both Sides

To isolate the term involving $r$, we need to subtract 14 from both sides of the equation.

$$-17 = 17r$$

Step 7: Divide Both Sides by 17

Finally, we can solve for $r$ by dividing both sides of the equation by 17.

$$r = -17/17$$

Conclusion

In this article, we have solved the linear equation $-3(1 + 6r) = 14 - r$ using step-by-step instructions. We have distributed the negative coefficient, simplified the equation, moved all terms to one side, combined like terms, and finally solved for $r$. By following these steps, we can solve any linear equation with ease.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When distributing a negative coefficient, remember to change the sign of each term inside the parentheses.
• When combining like terms, make sure to combine the coefficients of the same variables.
• When solving for a variable, make sure to isolate the variable on one side of the equation.

Real-World Applications

Linear equations have numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, linear equations can be used to model the motion of objects, while in engineering, they can be used to design and optimize systems. In economics, linear equations can be used to model the behavior of markets and economies.

Final Thoughts

Solving linear equations is a fundamental skill that is essential in various fields. By following the steps outlined in this article, you can solve any linear equation with ease. Remember to distribute the negative coefficient, simplify the equation, move all terms to one side, combine like terms, and finally solve for the variable. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It is a fundamental concept in mathematics and is used to model various real-world problems.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow the steps outlined in the article "Solving Linear Equations: A Step-by-Step Guide to Solving the Equation $-3(1 + 6r) = 14 - r$". This includes distributing the negative coefficient, simplifying the equation, moving all terms to one side, combining like terms, and finally solving for the variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. It stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I distribute a negative coefficient?

A: When distributing a negative coefficient, you need to change the sign of each term inside the parentheses. For example, if you have the equation $-3(1 + 6r)$, you would distribute the negative coefficient as follows:

$$-3(1 + 6r) = -3(1) - 3(6r)$$

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. For example, the equation $2x + 3 = 5$ is a linear equation, while the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it's essential to understand the steps involved in solving the equation and to use the calculator as a tool to check your work.

Q: How do I check my work when solving a linear equation?

A: To check your work when solving a linear equation, you can plug your solution back into the original equation and verify that it is true. For example, if you solve the equation $2x + 3 = 5$ and get $x = 1$, you can plug $x = 1$ back into the original equation to verify that it is true:

$$2(1) + 3 = 5$$

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Not following the order of operations (PEMDAS)
• Not distributing the negative coefficient correctly
• Not combining like terms correctly
• Not isolating the variable on one side of the equation
• Not checking your work

Q: Can I use linear equations to model real-world problems?

A: Yes, linear equations can be used to model various real-world problems. For example, in physics, linear equations can be used to model the motion of objects, while in engineering, they can be used to design and optimize systems. In economics, linear equations can be used to model the behavior of markets and economies.

Q: How do I apply linear equations to real-world problems?

A: To apply linear equations to real-world problems, you need to identify the variables and constants involved in the problem and set up a linear equation to model the situation. You can then use the equation to make predictions or solve for the unknown variable. For example, if you want to model the cost of producing a certain number of widgets, you can set up a linear equation to represent the cost and then use the equation to make predictions about the cost of producing different numbers of widgets.