Solve For \[$ Y \$\] In The Equation:$\[ 3x - 2y = 6 \\]

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Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving linear equations of the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Understanding the Given Equation

The given equation is 3x - 2y = 6. This is a linear equation in two variables, x and y. To solve for y, we need to isolate y on one side of the equation. In this case, we want to find the value of y, so we will need to get rid of the x term.

Isolating y

To isolate y, we can use the following steps:

  1. Add 2y to both sides of the equation: This will get rid of the -2y term on the left side of the equation.
  2. Divide both sides of the equation by -2: This will get rid of the -2 term on the left side of the equation.

Solving for y

Let's apply the steps above to the given equation:

3x - 2y = 6

Add 2y to both sides of the equation:

3x = 6 + 2y

Divide both sides of the equation by -2:

(3x) / -2 = (6 + 2y) / -2

x / -2 = 3 + y

Now, let's isolate y by subtracting 3 from both sides of the equation:

y = (x / -2) - 3

Simplifying the Expression

We can simplify the expression by multiplying both sides of the equation by -2:

-2y = -x - 6

Divide both sides of the equation by -2:

y = (x + 6) / 2

Conclusion

In this article, we have solved the linear equation 3x - 2y = 6 for y. We used the following steps:

  1. Add 2y to both sides of the equation.
  2. Divide both sides of the equation by -2.

By applying these steps, we were able to isolate y and find its value in terms of x.

Example Problems

Here are a few example problems to help you practice solving linear equations:

  1. Solve for y in the equation 2x + 3y = 5.
  2. Solve for y in the equation x - 4y = 2.
  3. Solve for y in the equation 5x - 2y = 3.

Tips and Tricks

Here are a few tips and tricks to help you solve linear equations:

  1. Use the distributive property: When multiplying a term by a binomial, use the distributive property to expand the expression.
  2. Combine like terms: When adding or subtracting terms, combine like terms to simplify the expression.
  3. Check your work: When solving a linear equation, check your work by plugging the solution back into the original equation.

Real-World Applications

Linear equations have many real-world applications, including:

  1. Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  2. Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  3. Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, solving linear equations is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve linear equations and find the value of y in terms of x. 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 ax + by = c. We also provided a step-by-step guide on how to isolate y and find its value in terms of x. In this article, we will answer some frequently asked questions about solving linear equations.

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 ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

  • The equation is in the form ax + by = c.
  • The highest power of the variable(s) is 1.
  • The equation does not contain any squared or cubed terms.

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. For example, the equation x + 2y = 3 is a linear equation, while the equation x^2 + 2y = 3 is a quadratic equation.

Q: How do I solve a linear equation with two variables?

A: To solve a linear equation with two variables, you need to isolate one of the variables. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: When solving a linear equation, follow the order of operations:

  1. Evaluate any expressions inside parentheses.
  2. Exponentiate any terms with exponents.
  3. Multiply and divide from left to right.
  4. Add and subtract from left to right.

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, make sure to check your work by plugging the solution back into the original 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 one of the variables.
  • Not checking your work.
  • Not using the correct signs when adding or subtracting terms.

Q: How do I check my work when solving a linear equation?

A: To check your work, plug the solution back into the original equation and simplify. If the equation is true, then your solution is correct.

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 model the motion of objects under constant acceleration, or to design and optimize systems.

Q: What are some real-world applications of linear equations?

A: Some real-world applications of linear equations include:

  • 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.

Conclusion

In conclusion, solving linear equations is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve linear equations and find the value of y in terms of x. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.