Solve For \[$ B \$\]:$\[ 9b - 9 = 90 \\]
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, 9b - 9 = 90, and provide a step-by-step guide on how to isolate the variable b.
Understanding the Equation
Before we dive into solving the equation, let's break it down and understand what it represents. The equation 9b - 9 = 90 is a linear equation in one variable, where b is the variable we want to solve for. The equation is in the form of ax + b = c, where a, b, and c are constants.
Step 1: Add 9 to Both Sides
To isolate the variable b, we need to get rid of the constant term -9 on the left-hand side of the equation. We can do this by adding 9 to both sides of the equation. This will keep the equation balanced and ensure that we are not changing the value of the variable b.
9b - 9 + 9 = 90 + 9
This simplifies to:
9b = 99
Step 2: Divide Both Sides by 9
Now that we have isolated the variable b, we need to get rid of the coefficient 9 that is multiplied by b. We can do this by dividing both sides of the equation by 9. This will give us the value of b.
(9b) / 9 = 99 / 9
This simplifies to:
b = 11
Conclusion
In this article, we have solved the linear equation 9b - 9 = 90 using a step-by-step approach. We added 9 to both sides of the equation to isolate the variable b, and then divided both sides by 9 to get the value of b. The final answer is b = 11.
Tips and Tricks
- When solving linear equations, it's essential to keep the equation balanced by performing the same operation on both sides.
- Use the order of operations (PEMDAS) to simplify the equation and make it easier to solve.
- Check your work by plugging the solution back into the original equation to ensure that it's true.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Common Mistakes
- Not keeping the equation balanced by performing the same operation on both sides.
- Not using the order of operations (PEMDAS) to simplify the equation.
- Not checking the solution by plugging it back into the original equation.
Conclusion
Introduction
In our previous article, we provided a step-by-step guide on how to solve the linear equation 9b - 9 = 90. In this article, we will answer some common questions that students and professionals may have when it comes to solving linear equations.
Q&A
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 of ax + b = c, where a, b, and c are 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(s) is 1, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I know if an equation is linear or quadratic?
A: To determine if an equation is linear or quadratic, look for the highest power of the variable(s). If the highest power is 1, it is a linear equation. If the highest power is 2, it is a quadratic equation.
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:
- 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 a linear equation with fractions?
A: To solve a linear equation with fractions, follow these steps:
- Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
- Simplify the equation by combining like terms.
- Solve for the variable using the steps outlined in our previous article.
Q: What is the difference between a linear equation and a system of linear equations?
A: A linear equation is a single equation in one variable, whereas a system of linear equations is a set of two or more linear equations in two or more variables. For example, 2x + 3 = 5 is a linear equation, while {2x + 3 = 5, x - 2y = 3} is a system of linear equations.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, follow these steps:
- Use the substitution method or the elimination method to eliminate one of the variables.
- Solve for the remaining variable using the steps outlined in our previous article.
- Substitute the value of the remaining variable back into one of the original equations to solve for the other variable.
Conclusion
Solving linear equations is a crucial skill that has numerous real-world applications. By understanding the basics of linear equations and following the steps outlined in this article, you can solve linear equations with ease. Remember to keep the equation balanced, use the order of operations (PEMDAS), and check your work to ensure that the solution is true.
Common Mistakes
- Not keeping the equation balanced by performing the same operation on both sides.
- Not using the order of operations (PEMDAS) to simplify the equation.
- Not checking the solution by plugging it back into the original equation.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Tips and Tricks
- Use the substitution method or the elimination method to solve a system of linear equations.
- Check your work by plugging the solution back into the original equation to ensure that it's true.
- Use the order of operations (PEMDAS) to simplify the equation and make it easier to solve.