15t+3t–12t–t+3t=16 What's The Answer?
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, 15t + 3t – 12t – t + 3t = 16, and provide a step-by-step guide on how to arrive at the solution.
Understanding the Equation
Before we dive into solving the equation, let's break it down and understand what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, t) is 1. The equation is also a combination of addition and subtraction operations.
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 raised to the same power. In this equation, we have several terms with the variable t:
- 15t
- 3t
- -12t
- -t
- 3t
To combine like terms, we need to add or subtract the coefficients of the terms with the same variable. In this case, we can combine the terms as follows:
15t + 3t - 12t - t + 3t = (15 + 3 - 12 - 1 + 3)t
Step 2: Simplify the Equation
Now that we have combined like terms, we can simplify the equation by evaluating the expression inside the parentheses:
(15 + 3 - 12 - 1 + 3)t = (8)t
So, the simplified equation is 8t = 16.
Step 3: Solve for t
To solve for t, we need to isolate the variable t on one side of the equation. We can do this by dividing both sides of the equation by 8:
8t = 16
t = 16/8
t = 2
Conclusion
In this article, we solved the linear equation 15t + 3t – 12t – t + 3t = 16 by combining like terms, simplifying the equation, and solving for t. The solution to the equation is t = 2.
Tips and Tricks
- When solving linear equations, it's essential to combine like terms to simplify the equation.
- Make sure to evaluate the expression inside the parentheses before simplifying the equation.
- When solving for t, make sure to isolate the variable t on one side of the 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 economic trends.
Common Mistakes
- Failing to combine like terms can lead to incorrect solutions.
- Not evaluating the expression inside the parentheses can lead to incorrect solutions.
- Not isolating the variable t on one side of the equation can lead to incorrect solutions.
Conclusion
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable is 1. It is a simple equation that can be solved using basic algebraic operations.
Q: What are like terms?
A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x 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 need to add or subtract the coefficients of the terms with the same variable. For example, if you have 2x + 4x, you can combine them by adding the coefficients: 2 + 4 = 6, so the combined term is 6x.
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 for a variable?
A: To solve for a variable, you need to isolate the variable on one side of the equation. This means getting the variable by itself on one side of the equation, without any other terms or constants.
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. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.
Q: Can I use a calculator to solve linear equations?
A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your work by plugging the solution back into the original equation to make sure it's true.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Failing to combine like terms
- Not evaluating expressions inside parentheses
- Not isolating the variable on one side of the equation
- Making errors when adding or subtracting numbers
Q: How can I practice solving linear equations?
A: There are many ways to practice solving linear equations, including:
- Working through practice problems in a textbook or online resource
- Using online tools or apps to generate practice problems
- Asking a teacher or tutor for help
- Joining a study group or math club to work with others
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 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 economic 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. Remember to combine like terms, simplify the equation, and solve for t to arrive at the correct solution.