Solve For { X $} . . . { -4 - 6x = -3x - 5x \}
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 linear equations, specifically the equation -4 - 6x = -3x - 5x. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants. The goal of solving a linear equation is to isolate the variable (x) on one side of the equation.
The Equation -4 - 6x = -3x - 5x
The equation we will be solving is -4 - 6x = -3x - 5x. This equation can be rewritten as -4 - 6x = -8x. Our goal is to isolate the variable 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 with the variable x: -6x and -3x. We can combine these terms by adding their coefficients (the numbers in front of the variable). -6x + (-3x) = -9x.
# Combine like terms
-6x + (-3x) = -9x
Step 2: Add 4 to Both Sides
The next step is to add 4 to both sides of the equation. This will help us get rid of the constant term (-4) on the left side of the equation. -4 + 4 = 0, so we can add 4 to both sides of the equation.
# Add 4 to both sides
-4 - 6x = -3x - 5x
4 + 4 = 8
-4 - 6x + 4 = -3x - 5x + 8
-2 - 6x = -8x + 8
Step 3: Add 6x to Both Sides
The next step is to add 6x to both sides of the equation. This will help us get rid of the term (-6x) on the left side of the equation.
# Add 6x to both sides
-2 - 6x + 6x = -8x + 8 + 6x
-2 = -2x + 8
Step 4: Subtract 8 from Both Sides
The next step is to subtract 8 from both sides of the equation. This will help us get rid of the constant term (8) on the right side of the equation.
# Subtract 8 from both sides
-2 - 8 = -2x + 8 - 8
-10 = -2x
Step 5: Divide Both Sides by -2
The final step is to divide both sides of the equation by -2. This will help us isolate the variable x on the left side of the equation.
# Divide both sides by -2
-10 / -2 = -2x / -2
5 = x
Conclusion
In this article, we solved the linear equation -4 - 6x = -3x - 5x using a step-by-step approach. We combined like terms, added 4 to both sides, added 6x to both sides, subtracted 8 from both sides, and finally divided both sides by -2 to isolate the variable x. The solution to the equation is x = 5.
Tips and Tricks
- When solving linear equations, it's essential to follow the order of operations (PEMDAS).
- Combine like terms to simplify the equation.
- Add or subtract the same value to both sides of the equation to isolate the variable.
- Divide both sides of the equation by the coefficient of the variable to solve for x.
Practice Problems
- Solve the linear equation 2x + 3 = 5x - 2.
- Solve the linear equation x - 2 = 3x + 1.
- Solve the linear equation 4x - 2 = 2x + 5.
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
Introduction
In our previous article, we explored the concept of solving linear equations and provided a step-by-step guide on how to solve the equation -4 - 6x = -3x - 5x. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will address some of the most 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 (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I know if an equation is linear?
A: To determine if an equation is linear, look for the highest power of the variable. If the highest power is 1, then the equation is linear. For example, the equation 2x + 3 = 5 is linear because the highest power of x is 1.
Q: What is the order of operations?
A: The order of operations is a set of rules that tells us 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 combine like terms?
A: To combine like terms, look for terms that have the same variable and coefficient. For example, in the equation 2x + 3x = 5, we can combine the like terms 2x and 3x by adding their coefficients: 2x + 3x = 5x.
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. For example, the equation 2x + 3 = 5 is a linear equation, while the equation x^2 + 2x + 1 = 0 is a quadratic equation.
Q: How do I solve a linear equation with fractions?
A: To solve a linear equation with fractions, follow these steps:
- Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
- Simplify the equation by canceling out any common factors.
- Solve the equation using the steps outlined in our previous article.
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 variables. For example, the equation 2x + 3 = 5 is a linear equation, while the system of equations:
2x + 3 = 5 x - 2 = 3
is a system of linear equations.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, follow these steps:
- Use the substitution method to solve one equation for one variable.
- Substitute the expression for the variable into the other equation.
- Solve the resulting equation for the other variable.
- Check your solution by plugging it back into both original equations.
Conclusion
In this article, we addressed some of the most frequently asked questions about solving linear equations. We hope that this Q&A guide has been helpful in clarifying any confusion and providing a better understanding of the concept of solving linear equations. Remember to practice regularly and seek help when needed to become proficient in solving linear equations.
Practice Problems
- Solve the linear equation 2x + 3 = 5.
- Solve the linear equation x - 2 = 3.
- Solve the system of linear equations:
2x + 3 = 5 x - 2 = 3
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
In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach and using the order of operations, we can solve linear equations with ease. Remember to combine like terms, add or subtract the same value to both sides of the equation, and divide both sides by the coefficient of the variable to solve for x. With practice and patience, you'll become a master of solving linear equations in no time!