What Is The Value Of $x$ In The Equation $2.5(6x - 4) = 10 + 4(1.5 + 0.5x)$?A. $ 1 2 \frac{1}{2} 2 1 [/tex] B. 2 C. $\frac{1}{3}$
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, $2.5(6x - 4) = 10 + 4(1.5 + 0.5x)$, and explore the value of $x$.
Understanding the Equation
The given equation is a linear equation, which means it can be written in the form $ax + b = c$. In this case, the equation is $2.5(6x - 4) = 10 + 4(1.5 + 0.5x)$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation.
Step 1: Distribute the Coefficients
The first step in solving the equation is to distribute the coefficients on both sides of the equation. This means multiplying the numbers outside the parentheses by the numbers inside the parentheses.
Simplifying the expressions, we get:
Step 2: Combine Like Terms
The next step is to combine like terms on both sides of the equation. This means adding or subtracting terms that have the same variable.
Subtracting $2x$ from both sides, we get:
Step 3: Add 10 to Both Sides
To isolate the term with $x$, we need to add 10 to both sides of the equation.
Simplifying the expressions, we get:
Step 4: Divide Both Sides by 13
Finally, we need to divide both sides of the equation by 13 to solve for $x$.
Simplifying the expressions, we get:
Conclusion
In this article, we solved a linear equation, $2.5(6x - 4) = 10 + 4(1.5 + 0.5x)$, and found the value of $x$ to be 2. We followed a step-by-step approach, distributing the coefficients, combining like terms, adding 10 to both sides, and finally dividing both sides by 13. This problem demonstrates the importance of following the order of operations and using algebraic techniques to solve linear equations.
Key Takeaways
- Linear equations can be solved using algebraic techniques.
- Distributing coefficients and combining like terms are essential steps in solving linear equations.
- Adding and subtracting terms with the same variable can help isolate the term with the variable.
- Dividing both sides of the equation by a coefficient can solve for the variable.
Practice Problems
Try solving the following linear equations:
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.
Conclusion
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 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 isolate the variable on one side of the equation. This can be done by using algebraic techniques such as distributing coefficients, combining like terms, adding and subtracting terms with the same variable, and dividing both sides of the equation by a coefficient.
Q: What is the order of operations in solving linear equations?
A: The order of operations in solving linear equations is:
- Distribute coefficients
- Combine like terms
- Add and subtract terms with the same variable
- Divide both sides of the equation by a coefficient
Q: How do I distribute coefficients in a linear equation?
A: To distribute coefficients in a linear equation, you need to multiply the numbers outside the parentheses by the numbers inside the parentheses. For example, in the equation $2(3x + 4)$, you would multiply 2 by 3x and 2 by 4.
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: Can I use a calculator to solve linear equations?
A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your work by plugging the solution back into the original equation.
Q: How do I check my work when solving a linear equation?
A: To check your work when solving a linear equation, you need to plug the solution back into the original equation and make sure that it is true. For example, if you solve the equation $2x + 3 = 5$ and get $x = 1$, you would plug $x = 1$ back into the original equation to make sure that it 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 coefficients correctly
- Not combining like terms correctly
- Not adding and subtracting terms with the same variable correctly
- Not dividing both sides of the equation by a coefficient correctly
Q: How do I practice solving linear equations?
A: To practice solving linear equations, you can try solving a variety of problems, such as:
- Solving linear equations with one variable
- Solving linear equations with two variables
- Solving linear equations with fractions and decimals
- Solving linear equations with negative numbers
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
In conclusion, solving linear equations is a crucial skill for students to master. By following a step-by-step approach and using algebraic techniques, we can solve linear equations and find the value of the variable. This article has provided a comprehensive guide to solving linear equations, including frequently asked questions and real-world applications.