Solve For { Q $} . . . { 2q + 18 = -5q - 3 \} Options:A. { Q = -5 $}$ B. { Q = -3 $}$ C. { Q = 3 $}$ D. { Q = 5 $}$
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 first-degree equation in one variable. We will use the given equation 2q + 18 = -5q - 3 as an example to demonstrate the step-by-step process of solving linear equations.
What are Linear Equations?
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 Given Equation
The given equation is 2q + 18 = -5q - 3. This equation is a linear equation in one variable, q. Our goal is to solve for q, which means we need to isolate q on one side of the equation.
Step 1: Add 5q to Both Sides
To solve for q, we need to get all the terms involving q on one side of the equation. We can do this by adding 5q to both sides of the equation. This will give us:
2q + 18 + 5q = -5q - 3 + 5q
Simplifying the equation, we get:
7q + 18 = -3
Step 2: Subtract 18 from Both Sides
Next, we need to get rid of the constant term on the left side of the equation. We can do this by subtracting 18 from both sides of the equation. This will give us:
7q + 18 - 18 = -3 - 18
Simplifying the equation, we get:
7q = -21
Step 3: Divide Both Sides by 7
Finally, we need to isolate q on one side of the equation. We can do this by dividing both sides of the equation by 7. This will give us:
7q / 7 = -21 / 7
Simplifying the equation, we get:
q = -3
Conclusion
In this article, we solved a linear equation using the step-by-step process of adding, subtracting, and dividing. We started with the given equation 2q + 18 = -5q - 3 and ended up with the solution q = -3. This demonstrates the importance of following the order of operations and using algebraic manipulation to solve linear equations.
Options
Now that we have solved the equation, let's look at the options provided:
A. q = -5 B. q = -3 C. q = 3 D. q = 5
Based on our solution, we can see that option B, q = -3, is the correct answer.
Tips and Tricks
Here are some tips and tricks to help you solve linear equations:
- Always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
- Use algebraic manipulation to isolate the variable on one side of the equation.
- Check your work by plugging the solution back into the original equation.
- Practice, practice, practice! The more you practice solving linear equations, the more comfortable you will become with the process.
Real-World Applications
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 future trends.
Conclusion
Introduction
In our previous article, we discussed the step-by-step process of solving linear equations. However, we know that practice makes perfect, and sometimes, it's helpful to have a Q&A guide to clarify any doubts or questions you may have. In this article, we will provide a Q&A guide to help you better understand how to solve 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 follow the step-by-step process of adding, subtracting, and dividing. Here's a general outline:
- Add or subtract the same value to both sides of the equation to isolate the variable.
- Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable.
- Check your work by plugging the solution back into the original equation.
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 you have multiple operations in an expression. 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 handle fractions in linear equations?
A: When working with fractions in linear equations, you can multiply both sides of the equation by the denominator to eliminate the fraction. For example, if you have the equation 1/2x + 3 = 5, you can multiply both sides by 2 to get x + 6 = 10.
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 ensure accuracy.
Q: What are some common mistakes to avoid when solving linear equations?
A: Here are some common mistakes to avoid when solving linear equations:
- Not following the order of operations.
- Not isolating the variable on one side of the equation.
- Not checking your work by plugging the solution back into the original equation.
- Not using algebraic manipulation to simplify the equation.
Q: How do I apply linear equations to real-world problems?
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 future trends.
Q: What are some tips for solving linear equations?
A: Here are some tips for solving linear equations:
- Practice, practice, practice! The more you practice solving linear equations, the more comfortable you will become with the process.
- Use algebraic manipulation to simplify the equation and isolate the variable.
- Check your work by plugging the solution back into the original equation.
- Use a calculator to check your work and ensure accuracy.
Conclusion
In conclusion, solving linear equations is an essential skill for students to master. By following the step-by-step process of adding, subtracting, and dividing, and by using algebraic manipulation to simplify the equation, you can solve linear equations and apply them to real-world problems. Remember to always follow the order of operations, check your work, and practice, practice, practice!