Solve For { X $} . . . { 3.18x + 2.6 + 0.9x + X = 28 \}
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, which involves combining like terms and isolating the variable. We will use the equation 3.18x + 2.6 + 0.9x + x = 28 as an example.
Understanding the Equation
Before we dive into solving the equation, let's break it down and understand what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, x) is 1. The equation is also a simple linear equation, which means it can be solved using basic algebraic operations.
Combining Like Terms
The first step in solving the equation is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, we have two terms with the variable x: 3.18x and 0.9x. We can combine these two terms by adding their coefficients (the numbers in front of the variable).
# Combine like terms
like_terms = 3.18x + 0.9x
combined_like_terms = (3.18 + 0.9)x
The combined like terms are (3.18 + 0.9)x, which simplifies to 4.08x.
Simplifying the Equation
Now that we have combined like terms, we can simplify the equation by combining the constant terms. The constant terms are the terms that do not have a variable. In this equation, we have two constant terms: 2.6 and x. We can combine these two terms by adding them.
# Combine constant terms
constant_terms = 2.6 + x
simplified_constant_terms = 2.6 + 1x
The simplified constant terms are 2.6 + 1x, which simplifies to 2.6 + x.
Isolating the Variable
Now that we have simplified the equation, we can isolate the variable by getting all the terms with the variable on one side of the equation. We can do this by subtracting 2.6 from both sides of the equation.
# Isolate the variable
equation = 4.08x + 2.6 + x = 28
isolated_variable = 4.08x + x = 28 - 2.6
The isolated variable is 4.08x + x = 25.4.
Solving for x
Now that we have isolated the variable, we can solve for x by combining the like terms on the left-hand side of the equation.
# Solve for x
like_terms = 4.08x + x
combined_like_terms = (4.08 + 1)x
simplified_equation = 5.08x = 25.4
The simplified equation is 5.08x = 25.4.
Final Step
The final step is to solve for x by dividing both sides of the equation by 5.08.
# Solve for x
x = 25.4 / 5.08
x = 5
The value of x is 5.
Conclusion
In this article, we solved a linear equation by combining like terms, simplifying the equation, isolating the variable, and solving for x. We used the equation 3.18x + 2.6 + 0.9x + x = 28 as an example. By following these steps, we were able to solve for x and find the value of the variable.
Tips and Tricks
- When solving linear equations, it's essential to combine like terms and simplify the equation.
- Isolating the variable is a crucial step in solving linear equations.
- When solving for x, make sure to divide both sides of the equation by the coefficient of the variable.
Real-World Applications
Linear equations have numerous 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.
Common Mistakes
- Failing to combine like terms and simplify the equation.
- Not isolating the variable.
- Dividing both sides of the equation by the wrong coefficient.
Conclusion
Introduction
In our previous article, we covered the basics of solving linear equations, including combining like terms, simplifying the equation, isolating the variable, and solving for x. In this article, we will provide a Q&A guide to help you better understand and apply the concepts.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It is a simple equation that can be solved using basic algebraic operations.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients (the numbers in front of the variable) of the terms with the same variable raised to the same power.
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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.
Q: How do I isolate the variable?
A: To isolate the variable, you need to get all the terms with the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Q: What is the coefficient of a term?
A: The coefficient of a term is the number in front of the variable. For example, in the term 3x, the coefficient is 3.
Q: How do I solve for x?
A: To solve for x, you need to isolate the variable by getting all the terms with the variable on one side of the equation and then dividing both sides of the equation by the coefficient of the variable.
Q: What is the difference between a linear equation and a system of linear equations?
A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more linear equations with the same variable.
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, elimination, or graphing to find the values of the variables that satisfy all the equations.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Failing to combine like terms and simplify the equation.
- Not isolating the variable.
- Dividing both sides of the equation by the wrong coefficient.
- Making errors when adding, subtracting, multiplying, or dividing both sides of the equation.
Q: How do I check my answer?
A: To check your answer, you need to plug the value of x back into the original equation and verify that it is true.
Q: What are some real-world applications of linear equations?
A: Linear equations have numerous 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
Solving linear equations is a fundamental skill that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can solve linear equations and apply them to various fields, including physics, engineering, and economics. Remember to combine like terms, simplify the equation, isolate the variable, and solve for x to ensure accurate results.