$\[ \begin{aligned} 6(4x - 3) &= 30 & & \text{Original Equation} \\ 24x - 18 &= 30 & & \text{Step 1} \\ 24x - 18 + 18 &= 30 + 18 & & \text{Step 2} \\ 24x &= 48 & & \text{Step 3} \\ \frac{24x}{24} &= \frac{48}{24} & & \text{Step 4} \\ x &= 2 & &
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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 linear equations, specifically the equation 6(4x - 3) = 30. We will break down the solution into manageable steps, making it easier for readers to understand and follow along.
The Original Equation
The original equation is 6(4x - 3) = 30. This equation involves a linear expression inside parentheses, which we need to simplify and isolate the variable x.
Step 1: Distribute the Coefficient
To simplify the equation, we need to distribute the coefficient 6 to the terms inside the parentheses. This means multiplying 6 by each term inside the parentheses.
6(4x - 3) = 6(4x) - 6(3)
Step 2: Simplify the Expression
Now that we have distributed the coefficient, we can simplify the expression by combining like terms.
24x - 18 = 30
Step 3: Add 18 to Both Sides
To isolate the variable x, we need to get rid of the constant term -18. We can do this by adding 18 to both sides of the equation.
24x - 18 + 18 = 30 + 18
Step 4: Simplify the Equation
Now that we have added 18 to both sides, we can simplify the equation by combining like terms.
24x = 48
Step 5: Divide Both Sides by 24
To isolate the variable x, we need to get rid of the coefficient 24. We can do this by dividing both sides of the equation by 24.
\frac{24x}{24} = \frac{48}{24}
Step 6: Simplify the Equation
Now that we have divided both sides by 24, we can simplify the equation by canceling out the common factor.
x = 2
Conclusion
In this article, we have solved the linear equation 6(4x - 3) = 30 using a step-by-step approach. We have distributed the coefficient, simplified the expression, added 18 to both sides, and finally divided both sides by 24 to isolate the variable x. The final solution is x = 2.
Tips and Tricks
- When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are simplifying the equation correctly.
- When distributing a coefficient, make sure to multiply it by each term inside the parentheses.
- When adding or subtracting a constant term, make sure to add or subtract it from both sides of the equation.
- When dividing both sides by a coefficient, make sure to divide both sides by the coefficient.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
- Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and structural systems.
- Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and resource allocation.
Final Thoughts
Solving linear equations is a fundamental skill that is essential for success in mathematics and other fields. By following the steps outlined in this article, you can master the art of solving linear equations and apply it to real-world problems. Remember to always follow the order of operations, distribute coefficients correctly, and isolate variables to find the solution.
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Introduction
In our previous article, we covered the basics of solving linear equations, including distributing coefficients, simplifying expressions, and isolating variables. However, we know that practice makes perfect, and there's no better way to practice than by answering questions. In this article, we'll provide a Q&A guide to help you master the art of solving linear equations.
Q&A
Q: What is the first step in solving a linear equation?
A: The first step in solving a linear equation is to simplify the expression by distributing coefficients and combining like terms.
Q: How do I distribute a coefficient to a term inside parentheses?
A: To distribute a coefficient to a term inside parentheses, you multiply the coefficient by each term inside the parentheses.
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, whereas a quadratic equation is an equation in which the highest power of the variable is 2.
Q: How do I isolate a variable in a linear equation?
A: To isolate a variable in a linear equation, you need to get rid of the constant term by adding or subtracting it from both sides of the equation.
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a mnemonic device that helps you remember the order in which to perform mathematical operations:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any 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 solve a linear equation with fractions?
A: To solve a linear equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
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, whereas a system of linear equations is a set of two or more equations with two or more variables.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to find the values of the variables that satisfy all the equations in the system. You can use substitution or elimination methods to solve a system of linear equations.
Tips and Tricks
- When solving linear equations, make sure to follow the order of operations (PEMDAS) to ensure that you are simplifying the equation correctly.
- When distributing a coefficient, make sure to multiply it by each term inside the parentheses.
- When adding or subtracting a constant term, make sure to add or subtract it from both sides of the equation.
- When dividing both sides by a coefficient, make sure to divide both sides by the coefficient.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
- Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and structural systems.
- Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and resource allocation.
Final Thoughts
Solving linear equations is a fundamental skill that is essential for success in mathematics and other fields. By following the steps outlined in this article and practicing with the Q&A guide, you can master the art of solving linear equations and apply it to real-world problems. Remember to always follow the order of operations, distribute coefficients correctly, and isolate variables to find the solution.
Additional Resources
For more practice and review, we recommend the following resources:
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
Conclusion
In this article, we provided a Q&A guide to help you master the art of solving linear equations. We covered topics such as distributing coefficients, simplifying expressions, and isolating variables. We also provided tips and tricks for solving linear equations and discussed real-world applications. By following the steps outlined in this article and practicing with the Q&A guide, you can become proficient in solving linear equations and apply it to real-world problems.