Solve The Equation $1 - 9x = 6x + 9$ Algebraically. Answer As A reduced Proper Or Improper Fraction Only.$x = \square$
Introduction
Linear equations are a fundamental concept in algebra, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $1 - 9x = 6x + 9$, algebraically. We will break down the solution into manageable steps, using a combination of mathematical operations and algebraic techniques.
Understanding the Equation
Before we dive into the solution, let's take a closer look at the equation. The equation is a linear equation in one variable, x. The left-hand side of the equation consists of a constant term, 1, and a term involving the variable x, -9x. The right-hand side of the equation consists of a term involving the variable x, 6x, and a constant term, 9.
Step 1: Add 9x to Both Sides
To solve the equation, we need to isolate the variable x. The first step is to add 9x to both sides of the equation, which will eliminate the term -9x from the left-hand side.
Simplifying the left-hand side, we get:
Step 2: Combine Like Terms
The next step is to combine like terms on the right-hand side of the equation. We can combine the terms 9x and 6x, as well as the constant terms 9 and 1.
Step 3: Subtract 9 from Both Sides
To isolate the term involving x, we need to subtract 9 from both sides of the equation.
Simplifying the left-hand side, we get:
Step 4: Divide Both Sides by 15
Finally, we need to divide both sides of the equation by 15 to solve for x.
Simplifying the right-hand side, we get:
Conclusion
In this article, we solved the linear equation $1 - 9x = 6x + 9$ algebraically. We broke down the solution into manageable steps, using a combination of mathematical operations and algebraic techniques. The final solution is:
This is a reduced proper fraction, which is the required format for the solution.
Tips and Tricks
When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to combine like terms whenever possible. Additionally, make sure to check your work by plugging the solution back into the original equation.
Common Mistakes
When solving linear equations, some common mistakes to avoid include:
- Not following the order of operations (PEMDAS)
- Not combining like terms
- Not checking the solution by plugging it back into the original equation
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 solved the linear equation $1 - 9x = 6x + 9$ algebraically. In this article, we will answer some common questions that students often have when solving linear equations.
Q: What is a linear equation?
A linear equation is an equation in which the highest power of the variable (x) 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.
Q: How do I solve a linear equation?
To solve a linear equation, you need to isolate the variable (x) on one side of the equation. You can do this by using inverse operations, such as addition, subtraction, multiplication, and division.
Q: What are inverse operations?
Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division.
Q: How do I use inverse operations to solve a linear equation?
To use inverse operations to solve a linear equation, you need to follow these steps:
- Identify the variable (x) and the constant terms on both sides of the equation.
- Use inverse operations to isolate the variable (x) on one side of the equation.
- Simplify the equation by combining like terms.
Q: What is a like term?
A like term is a term that has the same variable (x) and the same exponent. For example, 2x and 4x are like terms, as are 3 and 9.
Q: How do I combine like terms?
To combine like terms, you need to add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x, and 3 + 9 = 12.
Q: What is a coefficient?
A coefficient is a number that is multiplied by a variable (x). For example, in the equation 2x, the coefficient is 2.
Q: How do I check my work when solving a linear equation?
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.
Q: What are some common mistakes to avoid when solving linear equations?
Some common mistakes to avoid when solving linear equations include:
- Not following the order of operations (PEMDAS)
- Not combining like terms
- Not checking the solution by plugging it back into the original equation
Q: What are some real-world applications of linear equations?
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 the steps outlined in this article, you can solve linear equations algebraically and gain a deeper understanding of the underlying mathematical concepts. Remember to always follow the order of operations (PEMDAS), combine like terms, and check your work by plugging the solution back into the original equation.
Additional Resources
For additional resources on solving linear equations, including video tutorials and practice problems, please visit the following websites:
- Khan Academy: www.khanacademy.org
- Mathway: www.mathway.com
- IXL: www.ixl.com
FAQs
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 (x) is 1, while a quadratic equation is an equation in which the highest power of the variable (x) is 2.
Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you need to use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.
Q: What is the quadratic formula? A: The quadratic formula is a formula that is used to solve quadratic equations. It is x = (-b ± √(b^2 - 4ac)) / 2a.
Q: How do I use the quadratic formula to solve a quadratic equation? A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of a, b, and c into the formula and simplify.