Solve For { X $} . . . { X + 4x = 24 \} Express Your Answer In Decimal Form.

Mar 7, 2025 by ADMIN 77 views

**Solving Linear Equations: A Step-by-Step Guide**

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 simple linear equation, $x + 4x = 24$ , and express the solution in decimal form. We will break down the solution into step-by-step instructions, making it easy to follow and understand.

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 Solve

The equation we will be solving is $x + 4x = 24$ . This equation can be simplified by combining like terms, which will make it easier to solve.

Step 1: Combine Like Terms

To combine like terms, we need to identify the terms that have the same variable and coefficient. In this case, the terms $x$ and $4x$ have the same variable, $x$ , but different coefficients. We can combine these terms by adding their coefficients.

# Combine like terms
x = 1
four_x = 4 * x
combined_terms = x + four_x
print(combined_terms)

The output of the code above will be 5x, which is the result of combining the like terms.

Step 2: Simplify the Equation

Now that we have combined the like terms, we can simplify the equation by rewriting it in a more manageable form. We can do this by dividing both sides of the equation by the coefficient of the variable.

# Simplify the equation
simplified_equation = 24 / 5
print(simplified_equation)

The output of the code above will be 4.8, which is the result of simplifying the equation.

Step 3: Solve for x

Now that we have simplified the equation, we can solve for $x$ by dividing both sides of the equation by the coefficient of the variable.

# Solve for x
x = simplified_equation
print(x)

The output of the code above will be 4.8, which is the solution to the equation.

Conclusion

In this article, we have solved a simple linear equation, $x + 4x = 24$ , and expressed the solution in decimal form. We have broken down the solution into step-by-step instructions, making it easy to follow and understand. By combining like terms, simplifying the equation, and solving for $x$ , we have arrived at the solution, $x = 4.8$ . This solution can be verified by plugging it back into the original equation.

Tips and Variations

To solve a linear equation with a variable on both sides, we can use the method of substitution or elimination.
To solve a linear equation with a fraction, we can multiply both sides of the equation by the denominator to eliminate the fraction.
To solve a linear equation with a negative coefficient, we can multiply both sides of the equation by -1 to eliminate the negative sign.

Practice Problems

Solve the equation $2x + 3x = 12$ .
Solve the equation $x - 2x = 8$ .
Solve the equation $3x + 2x = 15$ .

References

Introduction

In our previous article, we discussed how to solve a simple linear equation, $x + 4x = 24$ , and expressed the solution in decimal form. In this article, we will provide a Q&A guide to help you understand and apply the concepts of 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 $ax + b = c$ , 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 can follow these steps:

Combine like terms: Combine the terms that have the same variable and coefficient.
Simplify the equation: Rewrite the equation in a more manageable form by dividing both sides of the equation by the coefficient of the variable.
Solve for x: Divide both sides of the equation by the coefficient of the variable to solve for x.

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, $x + 4x = 24$ is a linear equation, while $x^2 + 4x = 24$ is a quadratic equation.

Q: How do I solve a linear equation with a variable on both sides?

A: To solve a linear equation with a variable on both sides, you can use the method of substitution or elimination. For example, if you have the equation $x + 2x = 12$ , you can add the two terms on the left-hand side to get $3x = 12$ , and then divide both sides by 3 to get $x = 4$ .

Q: How do I solve a linear equation with a fraction?

A: To solve a linear equation with a fraction, you can multiply both sides of the equation by the denominator to eliminate the fraction. For example, if you have the equation $\frac{x}{2} + 3 = 12$ , you can multiply both sides by 2 to get $x + 6 = 24$ , and then subtract 6 from both sides to get $x = 18$ .

Q: How do I solve a linear equation with a negative coefficient?

A: To solve a linear equation with a negative coefficient, you can multiply both sides of the equation by -1 to eliminate the negative sign. For example, if you have the equation $-2x + 3 = 12$ , you can multiply both sides by -1 to get $2x - 3 = -12$ , and then add 3 to both sides to get $2x = -9$ , and finally divide both sides by 2 to get $x = -\frac{9}{2}$ .

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

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

Not combining like terms
Not simplifying the equation
Not solving for x
Not checking the solution by plugging it back into the original equation

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through examples and exercises in a textbook or online resource. You can also try solving linear equations on your own by creating your own problems and solutions.

Conclusion

Solving linear equations is an important skill to master in mathematics. By following the steps outlined in this article, you can solve linear equations with ease and confidence. Remember to combine like terms, simplify the equation, and solve for x to get the solution. With practice and patience, you can become proficient in solving linear equations and apply this skill to a wide range of mathematical problems.

Tips and Variations

To solve a linear equation with a variable on both sides, use the method of substitution or elimination.
To solve a linear equation with a fraction, multiply both sides of the equation by the denominator.
To solve a linear equation with a negative coefficient, multiply both sides of the equation by -1.
To practice solving linear equations, work through examples and exercises in a textbook or online resource.

Practice Problems

Solve the equation $2x + 3x = 12$ .
Solve the equation $x - 2x = 8$ .
Solve the equation $3x + 2x = 15$ .