{ \begin{tabular}{c|c} $x$ & $y = \frac{4}{3}x - 12$ \\ \hline 3 & $\square$ \\ \hline 6 & $\square$ \\ \hline 9 & $\square$ \\ \end{tabular} \}$
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 linear equations in the form of y = mx + b, where m is the slope and b is the y-intercept. We will use a specific equation, y = (4/3)x - 12, and substitute different values of x to find the corresponding values of y.
Understanding the Equation
The given equation is y = (4/3)x - 12. This equation represents a straight line with a slope of 4/3 and a y-intercept of -12. To solve for y, we need to isolate y on one side of the equation.
Substituting Values of x
We are given three values of x: 3, 6, and 9. We will substitute each of these values into the equation to find the corresponding values of y.
Substituting x = 3
To find the value of y when x = 3, we substitute x = 3 into the equation:
y = (4/3)(3) - 12
Using the order of operations, we first multiply 4/3 by 3:
(4/3)(3) = 4
Then, we subtract 12 from 4:
y = 4 - 12 y = -8
So, when x = 3, y = -8.
Substituting x = 6
To find the value of y when x = 6, we substitute x = 6 into the equation:
y = (4/3)(6) - 12
Using the order of operations, we first multiply 4/3 by 6:
(4/3)(6) = 8
Then, we subtract 12 from 8:
y = 8 - 12 y = -4
So, when x = 6, y = -4.
Substituting x = 9
To find the value of y when x = 9, we substitute x = 9 into the equation:
y = (4/3)(9) - 12
Using the order of operations, we first multiply 4/3 by 9:
(4/3)(9) = 12
Then, we subtract 12 from 12:
y = 12 - 12 y = 0
So, when x = 9, y = 0.
Conclusion
In this article, we solved a linear equation in the form of y = mx + b by substituting different values of x to find the corresponding values of y. We used the equation y = (4/3)x - 12 and substituted x = 3, 6, and 9 to find the values of y. By following the order of operations and isolating y on one side of the equation, we were able to find the values of y for each value of x.
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.
- Computer Science: Linear equations are used in algorithms and data structures to solve problems efficiently.
Tips and Tricks
Here are some tips and tricks for solving linear equations:
- Use the order of operations: When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you're performing the calculations correctly.
- Isolate y: To solve for y, you need to isolate y on one side of the equation. This can be done by adding or subtracting the same value to both sides of the equation.
- Use substitution: Substitution is a powerful technique for solving linear equations. By substituting different values of x, you can find the corresponding values of y.
Practice Problems
Here are some practice problems to help you master solving linear equations:
- Problem 1: Solve the equation y = 2x + 5 for x when y = 11.
- Problem 2: Solve the equation y = -3x + 2 for x when y = -5.
- Problem 3: Solve the equation y = (1/2)x - 3 for x when y = 2.
Conclusion
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 provide a Q&A guide to help you understand and solve linear equations. Whether you're a student, teacher, or simply looking to brush up on your math skills, this 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 (usually x) is 1. It can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate y on one side of the equation. This can be done by adding or subtracting the same value to both sides of the equation. You can also use substitution to solve for y.
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 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 use substitution to solve a linear equation?
A: Substitution is a powerful technique for solving linear equations. To use substitution, you need to substitute a value for x into the equation and solve for y. For example, if you have the equation y = 2x + 5 and you want to find the value of y when x = 3, you would substitute x = 3 into the equation and solve for y.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is useful for graphing linear equations and finding the equation of a line.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to plot two points on a coordinate plane and draw a line through them. You can also use the slope-intercept form of the equation to graph the line.
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: Make sure to follow the order of operations when solving an equation.
- Not isolating y: Make sure to isolate y on one side of the equation.
- Not using substitution: Substitution is a powerful technique for solving linear equations. Make sure to use it when necessary.
Q: How can I practice solving linear equations?
A: There are many ways to practice solving linear equations, including:
- Using online resources: There are many online resources available that provide practice problems and exercises for solving linear equations.
- Working with a tutor: Working with a tutor can be a great way to practice solving linear equations and get feedback on your work.
- Solving problems on your own: Solving problems on your own is a great way to practice solving linear equations and build your skills and confidence.
Conclusion
Solving linear equations is a crucial skill for students to master. By following the order of operations, isolating y, and using substitution, you can solve linear equations with ease. Remember to practice regularly to build your skills and confidence. With practice and patience, you'll become a master of solving linear equations in no time!