Solve For $w$.$7w - 5w = 10$Simplify Your Answer As Much As Possible.$ W = W = W = [/tex]
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 type of linear equation, which is a simple equation with one variable. We will use the equation $7w - 5w = 10$ as an example to demonstrate the step-by-step process of solving for the variable $w$.
What is a Linear Equation?
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. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.
The Equation to Solve
The equation we will be solving is $7w - 5w = 10$. This equation is a simple linear equation with one variable, $w$. Our goal is to isolate the variable $w$ and find its value.
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 $w$: $7w$ and $-5w$. We can combine these terms by adding them together.
So, the equation becomes $2w = 10$.
Step 2: Divide Both Sides by 2
Now that we have combined like terms, we can isolate the variable $w$ by dividing both sides of the equation by 2.
This simplifies to $w = 5$.
Conclusion
In this article, we have solved a simple linear equation using the equation $7w - 5w = 10$. We combined like terms, isolated the variable $w$, and found its value. The final answer is $w = 5$.
Why is Solving Linear Equations Important?
Solving linear equations is an essential skill in mathematics, and it has many real-world applications. For example, in physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the behavior of markets. In computer science, linear equations are used to solve systems of equations and optimize algorithms.
Tips and Tricks for Solving Linear Equations
Here are some tips and tricks for solving linear equations:
- Combine like terms: When solving a linear equation, combine like terms to simplify the equation.
- Isolate the variable: Isolate the variable by dividing both sides of the equation by a coefficient or adding/subtracting the same value to both sides.
- Check your work: Check your work by plugging the solution back into the original equation.
- Use algebraic manipulation: Use algebraic manipulation to simplify the equation and isolate the variable.
Common Mistakes to Avoid
Here are some common mistakes to avoid when solving linear equations:
- Not combining like terms: Failing to combine like terms can make the equation more complicated and difficult to solve.
- Not isolating the variable: Failing to isolate the variable can make it difficult to find the solution.
- Not checking your work: Failing to check your work can lead to incorrect solutions.
Real-World Applications of Solving Linear Equations
Solving linear equations has many real-world applications. Here are a few examples:
- Physics: Linear equations are used to describe the motion of objects. For example, the equation $s = ut + \frac{1}{2}at^2$ describes the position of an object as a function of time.
- Economics: Linear equations are used to model the behavior of markets. For example, the equation $P = MC + \frac{1}{2}t$ describes the price of a product as a function of its marginal cost and time.
- Computer Science: Linear equations are used to solve systems of equations and optimize algorithms. For example, the equation $Ax = b$ describes a system of linear equations, where $A$ is a matrix, $x$ is a vector of variables, and $b$ is a vector of constants.
Conclusion
Introduction
In our previous article, we discussed how to solve a simple linear equation using the equation $7w - 5w = 10$. We combined like terms, isolated the variable $w$, and found its value. The final answer was $w = 5$. In this article, we will answer some 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(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: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable by combining like terms, dividing both sides of the equation by a coefficient, or adding/subtracting the same value to both sides.
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. For example, the equation $x + 2 = 5$ is a linear equation, while the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.
Q: Can I use a calculator to solve a linear equation?
A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not combining like terms
- Not isolating the variable
- Not checking your work
- Using the wrong order of operations
Q: How do I check my work when solving a linear equation?
A: To check your work, plug the solution back into the original equation and see if it's true. For example, if you solved the equation $x + 2 = 5$ and got $x = 3$, you would plug $x = 3$ back into the original equation to get $3 + 2 = 5$, which is true.
Q: Can I use algebraic manipulation to solve a linear equation?
A: Yes, you can use algebraic manipulation to solve a linear equation. For example, you can use the distributive property to expand a product of two binomials, or you can use the commutative property to rearrange the terms in an equation.
Q: What are some real-world applications of solving linear equations?
A: Solving linear equations has many real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects.
- Economics: Linear equations are used to model the behavior of markets.
- Computer Science: Linear equations are used to solve systems of equations and optimize algorithms.
Conclusion
In this article, we have answered some frequently asked questions about solving linear equations. We have discussed what a linear equation is, how to solve a linear equation, and some common mistakes to avoid. We have also discussed some real-world applications of solving linear equations. By following the tips and tricks outlined in this article, you can improve your skills in solving linear equations and apply them to real-world problems.
Additional Resources
If you want to learn more about solving linear equations, here are some additional resources:
- Khan Academy: Solving Linear Equations
- Mathway: Solving Linear Equations
- Wolfram Alpha: Solving Linear Equations
Practice Problems
Here are some practice problems to help you improve your skills in solving linear equations:
- Solve the equation $2x + 3 = 7$.
- Solve the equation $x - 2 = 4$.
- Solve the equation $3x + 2 = 11$.
Answer Key
Here are the answers to the practice problems: