Solve $3a - 2(a + 1) = 15$.$a = $
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, namely the equation $3a - 2(a + 1) = 15$. We will break down the solution process into manageable steps, making it easy for readers to follow along and understand the concept.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, the equation is in the form of $ax + b = c$, 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 be Solved
The equation we will be solving is $3a - 2(a + 1) = 15$. This equation involves a variable $a$, and our goal is to isolate $a$ and find its value.
Step 1: Distribute the Negative 2
To begin solving the equation, we need to distribute the negative 2 to the terms inside the parentheses. This will give us $3a - 2a - 2 = 15$.
# Distributing the negative 2
equation = "3a - 2(a + 1) = 15"
distributed_equation = "3a - 2a - 2 = 15"
print(distributed_equation)
Step 2: Combine Like Terms
Next, we need to combine the like terms in the equation. In this case, we have two terms with the variable $a$, namely $3a$ and $-2a$. Combining these terms gives us $a - 2 = 15$.
# Combining like terms
distributed_equation = "3a - 2a - 2 = 15"
combined_equation = "a - 2 = 15"
print(combined_equation)
Step 3: Add 2 to Both Sides
To isolate the variable $a$, we need to get rid of the constant term $-2$ on the left-hand side of the equation. We can do this by adding 2 to both sides of the equation. This gives us $a = 17$.
# Adding 2 to both sides
combined_equation = "a - 2 = 15"
added_equation = "a = 17"
print(added_equation)
Conclusion
In this article, we solved the linear equation $3a - 2(a + 1) = 15$ using a step-by-step approach. We distributed the negative 2, combined like terms, and added 2 to both sides to isolate the variable $a$. The final solution is $a = 17$.
Tips and Tricks
- When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you're performing the operations in the correct order.
- When distributing a negative number, remember to change the sign of each term inside the parentheses.
- When combining like terms, make sure to combine the coefficients of the variable(s) and the constant terms separately.
Practice Problems
Try solving the following linear equations using the steps outlined in this article:
References
About the Author
Introduction
In our previous article, we solved the linear equation $3a - 2(a + 1) = 15$ using a step-by-step approach. In this article, we will answer some frequently asked questions about solving linear equations. Whether you're a student struggling to understand the concept or a teacher looking for ways to explain it to your students, this Q&A guide is for you.
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, the equation is in the form of $ax + b = c$, 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 follow these steps:
- Distribute any negative numbers or coefficients to the terms inside the parentheses.
- Combine like terms by adding or subtracting the coefficients of the variable(s) and the constant terms separately.
- Add or subtract the same value to both sides of the equation to isolate the variable.
- Check your solution by plugging it back into the original equation.
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 $x^2 + 4x + 4 = 0$ is a quadratic equation, while the equation $2x + 3 = 11$ is a linear 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 essential to understand the concept behind the solution and not just rely on the calculator. This will help you to solve more complex equations and understand the underlying mathematics.
Q: How do I check my solution to a linear equation?
A: To check your solution, plug it back into the original equation and see if it's true. If the equation holds true, then your solution is correct. If not, then you need to re-evaluate your solution.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not distributing negative numbers or coefficients correctly
- Not combining like terms correctly
- Not adding or subtracting the same value to both sides of the equation
- Not checking the solution by plugging it back into the original equation
Q: Can I use algebraic manipulation to solve linear equations?
A: Yes, you can use algebraic manipulation to solve linear equations. Algebraic manipulation involves using mathematical operations such as addition, subtraction, multiplication, and division to simplify the equation and isolate the variable.
Q: How do I graph linear equations?
A: To graph a linear equation, you need to plot two points on the coordinate plane that satisfy the equation. You can then draw a line through the two points to represent the equation.
Conclusion
Solving linear equations is a fundamental concept in mathematics, and understanding it is essential for success in algebra and beyond. By following the steps outlined in this Q&A guide, you'll be able to solve linear equations with confidence and accuracy. Remember to practice regularly and check your solutions to ensure that you're on the right track.
Practice Problems
Try solving the following linear equations using the steps outlined in this article:
References
About the Author
The author of this article is a mathematics educator with a passion for making complex concepts accessible to students of all levels. If you have any questions or comments, please don't hesitate to reach out.