Solve For { X$} . . . {5x = 2x + 12\}

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Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. In this article, we will focus on solving a simple linear equation, $5x = 2x + 12$, to demonstrate the step-by-step process involved in solving such equations.

Understanding the Equation

The given equation is $5x = 2x + 12$. This equation is a linear equation because it is in the form of $ax = bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = 5$, $b = 2$, and $c = 12$. The goal is to isolate the variable $x$ and find its value.

Step 1: Subtract $2x$ from Both Sides

To solve for $x$, we need to get all the terms involving $x$ on one side of the equation. We can do this by subtracting $2x$ from both sides of the equation. This will give us:

$$5x - 2x = 12$$

Step 2: Simplify the Equation

After subtracting $2x$ from both sides, we are left with:

$$3x = 12$$

Step 3: Divide Both Sides by 3

To isolate $x$, we need to get rid of the coefficient $3$ that is being multiplied by $x$. We can do this by dividing both sides of the equation by $3$. This will give us:

$$x = \frac{12}{3}$$

Step 4: Simplify the Fraction

The fraction $\frac{12}{3}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $3$. This will give us:

$$x = 4$$

Conclusion

In this article, we solved the linear equation $5x = 2x + 12$ by following the steps outlined above. We subtracted $2x$ from both sides, simplified the equation, divided both sides by $3$, and finally simplified the fraction to find the value of $x$. The final answer is $x = 4$.

Real-World Applications of Solving Linear Equations

Solving linear equations has numerous real-world applications in various fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations are used to describe the motion of objects, while in engineering, they are used to design and optimize systems. In economics, linear equations are used to model the behavior of markets and economies, and in computer science, they are used to develop algorithms and solve problems.

Tips and Tricks for Solving Linear Equations

Here are some tips and tricks for solving linear equations:

• Use inverse operations: To solve for $x$, use inverse operations to get rid of the coefficient being multiplied by $x$.
• Simplify the equation: Simplify the equation by combining like terms and canceling out common factors.
• Check your answer: Check your answer by plugging it back into the original equation to ensure that it is true.

Common Mistakes to Avoid When Solving Linear Equations

Here are some common mistakes to avoid when solving linear equations:

• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving linear equations.
• Not simplifying the equation: Simplify the equation by combining like terms and canceling out common factors.
• Not checking your answer: Check your answer by plugging it back into the original equation to ensure that it is true.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. By following the steps outlined in this article, you can solve linear equations and apply them to real-world problems. Remember to use inverse operations, simplify the equation, and check your answer to ensure that you are getting the correct solution.

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Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. In this article, we will focus on solving a simple linear equation, $5x = 2x + 12$, to demonstrate the step-by-step process involved in solving such equations.

Q&A: Solving Linear Equations

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable(s) on one side of the equation. You can do this by using inverse operations, such as addition, subtraction, multiplication, and division, to get rid of the coefficient being multiplied by the variable(s).

Q: What is the order of operations when solving linear equations?

A: The order of operations when solving linear equations 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 simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms and cancel out common factors. For example, if you have the equation $2x + 3x = 5x$, you can combine the like terms $2x$ and $3x$ to get $5x$.

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

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

A: To check your answer when solving a linear equation, you need to plug your solution back into the original equation to ensure 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 get $2(1) + 3 = 5$, which is true.

Real-World Applications of Solving Linear Equations

Solving linear equations has numerous real-world applications in various fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations are used to describe the motion of objects, while in engineering, they are used to design and optimize systems. In economics, linear equations are used to model the behavior of markets and economies, and in computer science, they are used to develop algorithms and solve problems.

Tips and Tricks for Solving Linear Equations

Here are some tips and tricks for solving linear equations:

• Use inverse operations: To solve for $x$, use inverse operations to get rid of the coefficient being multiplied by $x$.
• Simplify the equation: Simplify the equation by combining like terms and canceling out common factors.
• Check your answer: Check your answer by plugging it back into the original equation to ensure that it is true.

Common Mistakes to Avoid When Solving Linear Equations

Here are some common mistakes to avoid when solving linear equations:

• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving linear equations.
• Not simplifying the equation: Simplify the equation by combining like terms and canceling out common factors.
• Not checking your answer: Check your answer by plugging it back into the original equation to ensure that it is true.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. By following the steps outlined in this article, you can solve linear equations and apply them to real-world problems. Remember to use inverse operations, simplify the equation, and check your answer to ensure that you are getting the correct solution.