Solve For $y$.$\[ \begin{align*} 6x + 3y &= 21 \\ y &= [?]x + \square \end{align*} \\]
Introduction to Linear Equations
Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving linear equations of the form 6x + 3y = 21 to isolate y. We will use a step-by-step approach to simplify the equation and find the value of y.
Understanding the Equation
The given equation is 6x + 3y = 21. To isolate y, we need to get y by itself on one side of the equation. We can start by subtracting 6x from both sides of the equation.
Subtracting 6x from Both Sides
When we subtract 6x from both sides of the equation, we get:
6x + 3y - 6x = 21 - 6x
This simplifies to:
3y = 21 - 6x
Dividing Both Sides by 3
To isolate y, we need to get rid of the coefficient 3 that is multiplied by y. We can do this by dividing both sides of the equation by 3.
Dividing Both Sides by 3
When we divide both sides of the equation by 3, we get:
(3y) / 3 = (21 - 6x) / 3
This simplifies to:
y = (21 - 6x) / 3
Simplifying the Equation
We can simplify the equation further by dividing the numerator and denominator by their greatest common divisor, which is 3.
Simplifying the Equation
When we divide the numerator and denominator by 3, we get:
y = (7 - 2x) / 1
This simplifies to:
y = 7 - 2x
Conclusion
In this article, we have solved the linear equation 6x + 3y = 21 to isolate y. We have used a step-by-step approach to simplify the equation and find the value of y. The final equation is y = 7 - 2x, which is in the form y = [?]x + \square.
Real-World Applications
Linear equations have numerous real-world applications. For example, in physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the relationship between variables such as supply and demand. In engineering, linear equations are used to design and optimize systems.
Tips and Tricks
When solving linear equations, it is essential to follow the order of operations (PEMDAS):
- 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.
By following the order of operations, you can ensure that your calculations are accurate and that you arrive at the correct solution.
Common Mistakes to Avoid
When solving linear equations, there are several common mistakes to avoid:
- Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
- Not simplifying the equation: Failing to simplify the equation can make it difficult to isolate y.
- Not checking the solution: Failing to check the solution can lead to incorrect answers.
By avoiding these common mistakes, you can ensure that your solutions are accurate and that you arrive at the correct answer.
Practice Problems
To practice solving linear equations, try the following problems:
- Solve the equation 2x + 5y = 11 to isolate y.
- Solve the equation 3x - 2y = 7 to isolate y.
- Solve the equation x + 4y = 9 to isolate y.
By practicing these problems, you can improve your skills and become more confident in solving linear equations.
Conclusion
In conclusion, solving linear equations is a crucial skill in mathematics. By following a step-by-step approach and using the order of operations, you can isolate y and arrive at the correct solution. Remember to avoid common mistakes and practice regularly to improve your skills. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.
Introduction
In our previous article, we discussed how to solve linear equations of the form 6x + 3y = 21 to isolate y. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (x or y) is 1. For example, 2x + 3y = 5 is a linear equation.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable (x or y) by performing operations on both sides of the equation. You can add, subtract, multiply, or divide both sides of the equation to isolate the variable.
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 simplify an equation?
A: To simplify an equation, you need to combine like terms and eliminate any unnecessary operations. For example, if you have the equation 2x + 3x = 5, you can combine the like terms to get 5x = 5.
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 or y) is 1. A quadratic equation is an equation in which the highest power of the variable (x or y) is 2. For example, 2x + 3y = 5 is a linear equation, while x^2 + 2x + 1 = 0 is a quadratic equation.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to find the values of the variables that satisfy both equations. You can use substitution or elimination methods to solve a system of linear equations.
Q: What is the difference between a dependent and an independent variable?
A: A dependent variable is a variable that depends on the value of another variable. An independent variable is a variable that is not dependent on the value of another variable. For example, in the equation y = 2x, y is the dependent variable and x is the independent variable.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can then draw a line through the two points to represent the equation.
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 of the line and b is the y-intercept.
Q: How do I find the slope of a line?
A: To find the slope of a line, you need to use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.
Conclusion
In conclusion, solving linear equations is a crucial skill in mathematics. By following the order of operations and using the techniques discussed in this article, you can solve linear equations and apply them to real-world problems. Remember to practice regularly and seek help when you need it. With practice and patience, you can become proficient in solving linear equations and graphing lines.