Using Your Textbook, Solve The Problem Below. Show Your Work.Solve For X:${ X + 9.72 = 21 }$
Understanding the Problem
When solving a linear equation, the goal is to isolate the variable (in this case, x) on one side of the equation. The equation provided is:
To solve for x, we need to get rid of the constant term (+9.72) that is being added to x.
Step 1: Subtract 9.72 from Both Sides
To isolate x, we need to get rid of the +9.72 term. We can do this by subtracting 9.72 from both sides of the equation. This will keep the equation balanced.
Simplifying the Equation
When we subtract 9.72 from both sides, the +9.72 and -9.72 terms cancel each other out, leaving us with:
Performing the Subtraction
Now, we need to perform the subtraction on the right-hand side of the equation.
Conclusion
By following the steps outlined above, we have successfully solved for x in the equation:
The value of x is 11.28.
Why is this Important?
Solving linear equations is a fundamental concept in mathematics, and it has numerous applications in real-life situations. For example, in finance, linear equations can be used to calculate interest rates, investments, and loans. In science, linear equations can be used to model population growth, chemical reactions, and physical systems.
Tips and Tricks
When solving linear equations, it's essential to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
By following these steps and tips, you'll become more confident and proficient in solving linear equations.
Common Mistakes to Avoid
When solving linear equations, it's easy to make mistakes. Here are some common pitfalls to avoid:
- Not following the order of operations: Make sure to evaluate expressions inside parentheses first, followed by exponents, multiplication and division, and finally addition and subtraction.
- Not isolating the variable: Make sure to get the variable (x) on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation.
- Not checking your work: Double-check your calculations to ensure that you've solved the equation correctly.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Finance: Linear equations can be used to calculate interest rates, investments, and loans.
- Science: Linear equations can be used to model population growth, chemical reactions, and physical systems.
- Engineering: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Conclusion
Solving linear equations is a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined above and avoiding common mistakes, you'll become more confident and proficient in solving linear equations. Remember to always follow the order of operations and isolate the variable to ensure that you've solved the equation correctly.
Frequently Asked Questions
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (x) is 1. It can be written in the form:
ax + b = c
where a, b, and c are constants.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable (x) on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the 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 solving an equation. The order of operations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate 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 isolate the variable?
A: To isolate the variable, you need to get it on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the 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 (x) is 1, while a quadratic equation is an equation in which the highest power of the variable (x) is 2.
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 hand to make sure that you've solved the equation correctly.
Q: How do I check my work?
A: To check your work, you need to plug your solution back into the original equation and make sure that it's true. If it's not true, then you need to go back and re-solve the 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 following the order of operations
- Not isolating the variable
- Not checking your work
Q: Can I use linear equations to solve real-world problems?
A: Yes, linear equations can be used to solve real-world problems. For example, you can use linear equations to calculate interest rates, investments, and loans, or to model population growth, chemical reactions, and physical systems.
Q: How do I apply linear equations to real-world problems?
A: To apply linear equations to real-world problems, you need to identify the variables and constants in the problem, and then use the equation to solve for the unknown variable.
Q: What are some examples of real-world applications of linear equations?
A: Some examples of real-world applications of linear equations include:
- Finance: Calculating interest rates, investments, and loans
- Science: Modeling population growth, chemical reactions, and physical systems
- Engineering: Designing and optimizing systems, such as bridges, buildings, and electronic circuits
Q: Can I use linear equations to solve systems of equations?
A: Yes, you can use linear equations to solve systems of equations. A system of equations is a set of two or more equations that are all true at the same time.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to use a method such as substitution or elimination to find the values of the variables that satisfy all of the equations in the system.
Q: What are some common methods for solving systems of linear equations?
A: Some common methods for solving systems of linear equations include:
- Substitution: Solving one equation for one variable and then substituting that expression into the other equation
- Elimination: Adding or subtracting the equations in the system to eliminate one of the variables
- Graphing: Graphing the equations in the system on a coordinate plane and finding the point of intersection
Q: Can I use linear equations to solve optimization problems?
A: Yes, you can use linear equations to solve optimization problems. An optimization problem is a problem in which you need to find the maximum or minimum value of a function.
Q: How do I use linear equations to solve optimization problems?
A: To use linear equations to solve optimization problems, you need to identify the variables and constants in the problem, and then use the equation to find the maximum or minimum value of the function.
Q: What are some examples of optimization problems that can be solved using linear equations?
A: Some examples of optimization problems that can be solved using linear equations include:
- Finding the maximum or minimum value of a function
- Finding the optimal solution to a linear programming problem
- Finding the optimal solution to a network flow problem