Solve The Equation: 3 ( X − 5 ) = − 3 ( 2 X + 1 3(x - 5) = -3(2x + 1 3 ( X − 5 ) = − 3 ( 2 X + 1 ]

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 linear equation, $3(x - 5) = -3(2x + 1)$, and provide a step-by-step guide on how to approach it.

What are Linear Equations?

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 $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation to be Solved

The equation we will be solving is $3(x - 5) = -3(2x + 1)$. This equation involves parentheses, which can make it more challenging to solve. However, with the right approach, we can simplify the equation and solve for the variable $x$.

Step 1: Distribute the Numbers Outside the Parentheses

To solve the equation, we need to start by distributing the numbers outside the parentheses. This means multiplying the number outside the parentheses by each term inside the parentheses.

3(x - 5) = 3x - 15
-3(2x + 1) = -6x - 3

Step 2: Simplify the Equation

Now that we have distributed the numbers outside the parentheses, we can simplify the equation by combining like terms.

3x - 15 = -6x - 3

Step 3: Add 6x to Both Sides of the Equation

To isolate the variable $x$, we need to get all the terms with $x$ on one side of the equation. We can do this by adding $6x$ to both sides of the equation.

3x + 6x - 15 = -6x + 6x - 3
9x - 15 = -3

Step 4: Add 15 to Both Sides of the Equation

Next, we need to get rid of the constant term on the left side of the equation. We can do this by adding 15 to both sides of the equation.

9x - 15 + 15 = -3 + 15
9x = 12

Step 5: Divide Both Sides of the Equation by 9

Finally, we need to solve for the variable $x$. We can do this by dividing both sides of the equation by 9.

9x / 9 = 12 / 9
x = 4/3

Conclusion

Solving linear equations requires a step-by-step approach. By distributing the numbers outside the parentheses, simplifying the equation, adding or subtracting terms, and dividing both sides of the equation by the coefficient of the variable, we can solve for the variable $x$. In this article, we solved the equation $3(x - 5) = -3(2x + 1)$ and found that $x = 4/3$.

Tips and Tricks

• When solving linear equations, always start by distributing the numbers outside the parentheses.
• Simplify the equation by combining like terms.
• Add or subtract terms to isolate the variable.
• Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Common Mistakes to Avoid

• Not distributing the numbers outside the parentheses.
• Not simplifying the equation by combining like terms.
• Not adding or subtracting terms to isolate the variable.
• Not dividing both sides of the equation by the coefficient of the variable.

Real-World Applications

Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• 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 future trends.

Conclusion

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Introduction

In our previous article, we provided a step-by-step guide on how to solve linear equations. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will provide a Q&A guide on solving linear equations, covering common questions and scenarios that students may encounter.

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

1. Distribute the numbers outside the parentheses.
2. Simplify the equation by combining like terms.
3. Add or subtract terms to isolate the variable.
4. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

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, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I handle parentheses in a linear equation?

A: When handling parentheses in a linear equation, you need to distribute the numbers outside the parentheses to each term inside the parentheses. For example, in the equation $3(x - 5) = -3(2x + 1)$, you would distribute the 3 to each term inside the parentheses to get $3x - 15 = -6x - 3$.

Q: What is the order of operations in solving a linear equation?

A: The order of operations in solving a linear equation is:

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 check my solution to a linear equation?

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

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

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

• Not distributing the numbers outside the parentheses.
• Not simplifying the equation by combining like terms.
• Not adding or subtracting terms to isolate the variable.
• Not dividing both sides of the equation by the coefficient of the variable.

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

A: Linear equations can be applied to a wide range of real-world problems, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• 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 future trends.

Conclusion

Solving linear equations is a crucial skill for students to master. By following the step-by-step guide outlined in this article, students can solve linear equations with ease. Remember to distribute the numbers outside the parentheses, simplify the equation, add or subtract terms, and divide both sides of the equation by the coefficient of the variable to solve for the variable $x$.