Solve $4kx + 10kx = 7$ For $x$.A. $ X = 2 K X = 2k X = 2 K [/tex] B. $x = \frac{2}{k}$ C. $x = \frac{k}{2}$ D. $ X = 1 2 K X = \frac{1}{2k} X = 2 K 1 [/tex]
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, $4kx + 10kx = 7$, for the variable $x$. We will break down the solution step by step, using algebraic techniques to isolate the variable and find its value.
Understanding the Equation
The given equation is a linear equation in one variable, $x$. It involves a constant, $k$, and the variable $x$. The equation is:
Our goal is to solve for $x$, which means we need to isolate $x$ on one side of the equation.
Step 1: Combine Like Terms
The first step in solving the equation is to combine like terms. In this case, we have two terms involving $kx$, which can be combined as follows:
So, the equation becomes:
Step 2: Isolate the Variable
Now that we have combined like terms, we can isolate the variable $x$ by dividing both sides of the equation by $14k$. This will give us:
Step 3: Simplify the Expression
We can simplify the expression further by dividing both the numerator and the denominator by their greatest common divisor, which is 7. This gives us:
Conclusion
In this article, we solved the linear equation $4kx + 10kx = 7$ for the variable $x$. We broke down the solution into three steps: combining like terms, isolating the variable, and simplifying the expression. The final solution is:
This is the correct answer, which can be verified by plugging it back into the original equation.
Answer Key
The correct answer is:
This is option D in the multiple-choice question.
Tips and Tricks
When solving linear equations, it's essential to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
By following this order of operations, you can ensure that you solve the equation correctly and avoid any errors.
Common Mistakes
When solving linear equations, some common mistakes to avoid include:
- Not combining like terms
- Not isolating the variable
- Not simplifying the expression
- Not following the order of operations
By being aware of these common mistakes, you can avoid them and ensure that you solve the equation correctly.
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 economic trends.
By understanding how to solve linear equations, you can apply this knowledge to a wide range of real-world problems and make informed decisions.
Conclusion
Introduction
In our previous article, we solved the linear equation $4kx + 10kx = 7$ for the variable $x$. We broke down the solution into three steps: combining like terms, isolating the variable, and simplifying the expression. In this article, we will answer some frequently asked questions about 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 need to follow these steps:
- Combine like terms: Combine any terms that have the same variable(s) and coefficient(s).
- Isolate the variable: Get the variable(s) on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
- Simplify the expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary operations.
Q: What is the order of operations?
A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I handle fractions in linear equations?
A: When working with fractions in linear equations, you need to follow these steps:
- Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
- Simplify the resulting expression by combining like terms and eliminating any unnecessary operations.
Q: Can I use a calculator to solve linear equations?
A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the underlying math and be able to verify the solution using algebraic techniques.
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 isolating the variable
- Not simplifying the expression
- Not following the order of operations
- Not checking the solution by plugging it back into the original equation
Q: How do I apply linear equations to real-world problems?
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 economic trends.
By understanding how to solve linear equations, you can apply this knowledge to a wide range of real-world problems and make informed decisions.
Conclusion
In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear equations with confidence and apply this knowledge to real-world problems. Remember to combine like terms, isolate the variable, and simplify the expression, and always follow the order of operations. With practice and patience, you can become proficient in solving linear equations and tackle even the most challenging problems.