Solve The Equation:$\[-x + 8 + 3x = X - 6\\]Choose The Correct Solution:A. \[$x = -18\$\]B. \[$x = -14\$\]C. \[$x = 2\$\]D. \[$x = 4\$\]
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, step by step, and provide a clear explanation of the solution process. We will also discuss the importance of linear equations in real-world applications and provide examples of how they are used in various fields.
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.
The Equation to be Solved
The equation to be solved is:
-x + 8 + 3x = x - 6
This equation is a linear equation, and we will solve it step by step.
Step 1: Combine Like Terms
The first step in solving the equation is to combine like terms. Like terms are terms that have the same variable(s) raised to the same power.
In this equation, we have two like terms: -x and 3x. We can combine these terms by adding them together:
-x + 3x = 2x
So, the equation becomes:
2x + 8 = x - 6
Step 2: Isolate the Variable
The next step is to isolate the variable x. To do this, we need to get all the terms with x on one side of the equation and the constant terms on the other side.
We can start by subtracting x from both sides of the equation:
2x - x + 8 = x - x - 6
This simplifies to:
x + 8 = -6
Step 3: Solve for x
Now that we have isolated the variable x, we can solve for x by subtracting 8 from both sides of the equation:
x + 8 - 8 = -6 - 8
This simplifies to:
x = -14
Conclusion
The solution to the equation is x = -14. This means that the value of x that satisfies the equation is -14.
Importance of Linear Equations
Linear equations are used in a wide range of applications, including:
- Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of an object.
- Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and communication networks.
- Economics: Linear equations are used to model economic systems, including the supply and demand of goods and services.
- Computer Science: Linear equations are used in computer graphics, game development, and machine learning.
Real-World Examples
Here are a few examples of how linear equations are used in real-world applications:
- Designing a Bridge: A civil engineer uses linear equations to design a bridge, taking into account the weight of the bridge, the strength of the materials, and the wind and traffic loads.
- Optimizing a Supply Chain: A logistics manager uses linear equations to optimize the supply chain, taking into account the cost of transportation, storage, and inventory.
- Predicting Stock Prices: A financial analyst uses linear equations to predict stock prices, taking into account historical data, economic indicators, and market trends.
Conclusion
In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step guide outlined in this article, students can solve linear equations with confidence. Linear equations are used in a wide range of applications, including physics, engineering, economics, and computer science. By understanding how linear equations are used in real-world applications, students can appreciate the importance of mathematics in everyday life.
Final Answer
The final answer is:
Introduction
In our previous article, we discussed how to solve linear equations step by step. In this article, we will provide a Q&A guide to help students and professionals understand the concept of linear equations and how to solve them.
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: What are like terms?
A: Like terms are terms that have the same variable(s) raised to the same power. For example, in the equation 2x + 3x, 2x and 3x are like terms because they both have the variable x raised to the power of 1.
Q: How do I combine like terms?
A: To combine like terms, you add or subtract the coefficients of the like terms. For example, in the equation 2x + 3x, you can combine the like terms by adding the coefficients:
2x + 3x = 5x
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 and the constant terms on the other side. You can do this by adding or subtracting the same value to both sides of the equation.
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 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 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.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can then draw a line through the two points to represent the equation.
Q: What are some real-world applications of linear equations?
A: Linear equations have many real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of an object.
- Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and communication networks.
- Economics: Linear equations are used to model economic systems, including the supply and demand of goods and services.
- Computer Science: Linear equations are used in computer graphics, game development, and machine learning.
Conclusion
In conclusion, solving linear equations is a crucial skill for students and professionals to master. By understanding the concept of linear equations and how to solve them, you can apply this knowledge to a wide range of real-world applications. We hope this Q&A guide has been helpful in answering your questions and providing a better understanding of linear equations.
Final Tips
- Practice, practice, practice: The more you practice solving linear equations, the more comfortable you will become with the concept.
- Use online resources: There are many online resources available that can help you learn and practice solving linear equations, including video tutorials, practice problems, and interactive quizzes.
- Seek help when needed: Don't be afraid to ask for help if you are struggling with a particular concept or problem.