Solve The Equation:${ -9 + 8x = X + 1 - 10 }$
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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 a specific linear equation, step by step, and provide a clear understanding of the process involved.
What is a Linear Equation?
A linear equation is an algebraic equation in which the highest power of the variable(s) is 1. It can be written in the form:
ax + b = c
where a, b, and c are constants, and x is the variable.
The Given Equation
The equation we will be solving is:
-9 + 8x = x + 1 - 10
Step 1: Simplify the Equation
The first step in solving the equation is to simplify it by combining like terms. We can start by combining the constants on the right-hand side of the equation.
-9 + 8x = x + (1 - 10)
Combine the constants:
-9 + 8x = x - 9
Step 2: Add 9 to Both Sides
To isolate the variable x, we need to get rid of the constant term on the left-hand side of the equation. We can do this by adding 9 to both sides of the equation.
-9 + 9 + 8x = x - 9 + 9
This simplifies to:
8x = x
Step 3: Subtract x from Both Sides
To further isolate the variable x, we need to get rid of the x term on the right-hand side of the equation. We can do this by subtracting x from both sides of the equation.
8x - x = x - x
This simplifies to:
7x = 0
Step 4: Divide Both Sides by 7
Finally, to solve for x, we need to get rid of the coefficient of x, which is 7. We can do this by dividing both sides of the equation by 7.
(7x) / 7 = 0 / 7
This simplifies to:
x = 0
Conclusion
In this article, we have solved a linear equation step by step, using basic algebraic operations. We have shown that the solution to the equation is x = 0. This process can be applied to solve any linear equation, and it is an essential skill for students and professionals in mathematics and related fields.
Tips and Tricks
- When solving linear equations, it is essential to simplify the equation by combining like terms.
- To isolate the variable, add or subtract the same value to both sides of the equation.
- To get rid of a coefficient, divide both sides of the equation by that coefficient.
- Practice solving linear equations to become proficient in this skill.
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 computer networks.
- 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 has numerous applications in mathematics and related fields. By following the steps outlined in this article, you can become proficient in solving linear equations and apply this skill to real-world problems. Remember to practice regularly and to seek help when needed. With dedication and persistence, you can master the art of solving linear equations.
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Introduction
In our previous article, we covered the basics of solving linear equations, including simplifying the equation, isolating the variable, and getting rid of coefficients. In this article, we will answer some frequently asked questions about solving linear equations, providing additional insights and tips to help you master this skill.
Q&A
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, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, the equation 2x + 3 = 5 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I know if an equation is linear or quadratic?
A: To determine if an equation is linear or quadratic, look for the highest power of the variable(s). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.
Q: What is the order of operations when solving a linear equation?
A: When solving a linear equation, follow the order of operations (PEMDAS):
- 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 handle fractions when solving a linear equation?
A: When solving a linear equation with fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
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 linear equations with the same variables.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, use the following methods:
- Substitution: Substitute the expression for one variable from one equation into the other equation.
- Elimination: Add or subtract the equations to eliminate one variable.
- Graphing: Graph the equations on a coordinate plane and find the point of intersection.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not simplifying the equation before solving.
- Not isolating the variable correctly.
- Not getting rid of coefficients correctly.
- Not checking the solution for extraneous solutions.
Tips and Tricks
- Practice solving linear equations regularly to become proficient in this skill.
- Use the order of operations (PEMDAS) when solving linear equations.
- Eliminate fractions by multiplying both sides of the equation by the LCM of the denominators.
- Use substitution, elimination, or graphing to solve systems of linear equations.
- Check the solution for extraneous solutions.
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 computer networks.
- 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 has numerous applications in mathematics and related fields. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations and apply this skill to real-world problems. Remember to use the order of operations, eliminate fractions, and check the solution for extraneous solutions. With dedication and persistence, you can master the art of solving linear equations.