Think About The Equation $3y - 9 = 24$.a. The Value Of $3y$ Is 33. How Do You Know This Is True?
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 explore the process of solving linear equations, using the equation $3y - 9 = 24$ as a case study. We will also discuss how to verify the solution and provide a step-by-step guide on how to solve similar equations.
Understanding the Equation
The given equation is $3y - 9 = 24$. To solve for $y$, we need to isolate the variable $y$ on one side of the equation. The first step is to add $9$ to both sides of the equation to get rid of the negative term.
# Given equation
equation = "3y - 9 = 24"
# Add 9 to both sides of the equation
new_equation = "3y = 33"
Verifying the Solution
The value of $3y$ is given as $33$. To verify that this is true, we can substitute $y = 11$ into the original equation and check if it holds true.
# Substitute y = 11 into the original equation
y = 11
equation = "3y - 9 = 24"
result = 3 * y - 9
print(result) # Output: 33
As we can see, the result is indeed $33$, which confirms that the value of $3y$ is $33$.
Solving the Equation
Now that we have verified the solution, let's solve the equation $3y = 33$. To do this, we need to isolate the variable $y$ by dividing both sides of the equation by $3$.
# Divide both sides of the equation by 3
y = 33 / 3
print(y) # Output: 11
Therefore, the value of $y$ is $11$.
Step-by-Step Guide
Here is a step-by-step guide on how to solve linear equations like $3y - 9 = 24$:
- Add or subtract the same value to both sides of the equation: This will help to get rid of any negative terms and make the equation easier to solve.
- Multiply or divide both sides of the equation by the same value: This will help to isolate the variable and solve for its value.
- Verify the solution: Substitute the value of the variable into the original equation and check if it holds true.
Conclusion
Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve equations like $3y - 9 = 24$ and verify the solution. Remember to add or subtract the same value to both sides of the equation, multiply or divide both sides by the same value, and verify the solution to ensure that you have found the correct value of the variable.
Common Mistakes to Avoid
When solving linear equations, there are several common mistakes to avoid:
- Not adding or subtracting the same value to both sides of the equation: This can lead to incorrect solutions and make it difficult to verify the solution.
- Not multiplying or dividing both sides of the equation by the same value: This can also lead to incorrect solutions and make it difficult to verify the solution.
- Not verifying the solution: This is the most common mistake when solving linear equations. Make sure to substitute the value of the variable into the original equation and check if it holds true.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
- Economics: Linear equations are used to model economic systems and make predictions about future economic trends.
- Computer science: Linear equations are used in computer algorithms and data analysis.
Conclusion
Introduction
In our previous article, we explored the process of solving linear equations, using the equation $3y - 9 = 24$ as a case study. We also discussed how to verify the solution and provided a step-by-step guide on how to solve similar equations. In this article, we will answer some frequently asked questions about 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 in which the variable(s) are not raised to a power greater than 1.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable(s) on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides by the same value.
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.
Q: How do I verify the solution to a linear equation?
A: To verify the solution to a linear equation, you need to substitute the value of the variable(s) into the original equation and check if it holds true.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not adding or subtracting the same value to both sides of the equation
- Not multiplying or dividing both sides of the equation by the same value
- Not verifying the solution
Q: How do I use linear equations in real-world applications?
A: Linear equations have numerous real-world applications, including:
- Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
- Economics: Linear equations are used to model economic systems and make predictions about future economic trends.
- Computer science: Linear equations are used in computer algorithms and data analysis.
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 related to each other. You can use linear equations to solve systems of equations by using methods such as substitution or elimination.
Q: How do I use linear equations to solve inequalities?
A: You can use linear equations to solve inequalities by using methods such as substitution or elimination. Inequalities are equations in which the variable(s) are not equal to a value, but are greater than or less than a value.
Conclusion
In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve equations like $3y - 9 = 24$ and verify the solution. Remember to add or subtract the same value to both sides of the equation, multiply or divide both sides by the same value, and verify the solution to ensure that you have found the correct value of the variable.
Additional Resources
For more information on solving linear equations, check out the following resources:
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
Practice Problems
Try solving the following linear equations: