Solve For One Variable In The Equation:${ 2x - 3y = 12 }$

by ADMIN 59 views

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. 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 with two variables, where we need to find the value of one variable.

Understanding the Equation

The given equation is:

2x−3y=12{ 2x - 3y = 12 }

This is a linear equation with two variables, x and y. Our goal is to solve for one variable, either x or y.

Solving for One Variable

To solve for one variable, we need to isolate that variable on one side of the equation. We can do this by performing algebraic operations on the equation.

Solving for x

To solve for x, we need to isolate x on one side of the equation. We can do this by adding 3y to both sides of the equation:

2x−3y+3y=12+3y{ 2x - 3y + 3y = 12 + 3y }

This simplifies to:

2x=12+3y{ 2x = 12 + 3y }

Next, we can divide both sides of the equation by 2 to isolate x:

2x2=12+3y2{ \frac{2x}{2} = \frac{12 + 3y}{2} }

This simplifies to:

x=12+3y2{ x = \frac{12 + 3y}{2} }

Solving for y

To solve for y, we need to isolate y on one side of the equation. We can do this by subtracting 2x from both sides of the equation:

2x−3y−2x=12−2x{ 2x - 3y - 2x = 12 - 2x }

This simplifies to:

−3y=12−2x{ -3y = 12 - 2x }

Next, we can divide both sides of the equation by -3 to isolate y:

−3y−3=12−2x−3{ \frac{-3y}{-3} = \frac{12 - 2x}{-3} }

This simplifies to:

y=2x−123{ y = \frac{2x - 12}{3} }

Example Problem

Let's consider an example problem to illustrate the concept of solving linear equations.

Problem

Solve the equation:

4x+2y=20{ 4x + 2y = 20 }

Solution

To solve for x, we can add -2y to both sides of the equation:

4x+2y−2y=20−2y{ 4x + 2y - 2y = 20 - 2y }

This simplifies to:

4x=20−2y{ 4x = 20 - 2y }

Next, we can divide both sides of the equation by 4 to isolate x:

4x4=20−2y4{ \frac{4x}{4} = \frac{20 - 2y}{4} }

This simplifies to:

x=20−2y4{ x = \frac{20 - 2y}{4} }

To solve for y, we can subtract 4x from both sides of the equation:

4x+2y−4x=20−4x{ 4x + 2y - 4x = 20 - 4x }

This simplifies to:

2y=20−4x{ 2y = 20 - 4x }

Next, we can divide both sides of the equation by 2 to isolate y:

2y2=20−4x2{ \frac{2y}{2} = \frac{20 - 4x}{2} }

This simplifies to:

y=20−4x2{ y = \frac{20 - 4x}{2} }

Conclusion

In this article, we have discussed how to solve linear equations with two variables, where we need to find the value of one variable. We have used algebraic operations to isolate the variable on one side of the equation. We have also provided an example problem to illustrate the concept. By following the steps outlined in this article, you should be able to solve linear equations with two variables and find the value of one variable.

Tips and Tricks

  • Make sure to isolate the variable on one side of the equation.
  • Use algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable.
  • Check your solution by plugging it back into the original equation.

Common Mistakes

  • Failing to isolate the variable on one side of the equation.
  • Not using algebraic operations to isolate the variable.
  • Not checking the solution by plugging it back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under the influence of forces.
  • Engineering: Linear equations are used to design and optimize systems such as bridges, buildings, and electronic circuits.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Final Thoughts

Solving linear equations with two variables is a fundamental concept in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you should be able to solve linear equations with two variables and find the value of one variable. Remember to isolate the variable on one side of the equation, use algebraic operations to isolate the variable, and check your solution by plugging it back into the original equation.

Q: What is a linear equation with two variables?

A: A linear equation with two variables is an equation in which the highest power of the variable(s) is 1. It is a fundamental concept in mathematics and has numerous real-world applications.

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 variable on one side of the equation. You can do this by performing algebraic operations such as addition, subtraction, multiplication, and division.

Q: What are some common mistakes to avoid when solving linear equations with two variables?

A: Some common mistakes to avoid when solving linear equations with two variables include:

  • Failing to isolate the variable on one side of the equation.
  • Not using algebraic operations to isolate the variable.
  • Not checking the solution by plugging it back into the original equation.

Q: How do I check my solution to a linear equation with two variables?

A: To check your solution to a linear equation with two variables, you need to plug it back into the original equation. If the solution satisfies the equation, then it is correct.

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

A: Linear equations with two variables have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under the influence of forces.
  • Engineering: Linear equations are used to design and optimize systems such as bridges, buildings, and electronic circuits.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: Can I use a calculator to solve linear equations with two variables?

A: Yes, you can use a calculator to solve linear equations with two variables. However, it is always a good idea to check your solution by plugging it back into the original equation.

Q: How do I solve a linear equation with two variables when there are fractions involved?

A: To solve a linear equation with two variables when there are fractions involved, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple of the denominators.

Q: Can I solve a linear equation with two variables using a graphing calculator?

A: Yes, you can solve a linear equation with two variables using a graphing calculator. However, it is always a good idea to check your solution by plugging it back into the original equation.

Q: How do I solve a linear equation with two variables when there are decimals involved?

A: To solve a linear equation with two variables when there are decimals involved, you need to eliminate the decimals by multiplying both sides of the equation by a power of 10.

Q: Can I use a computer program to solve linear equations with two variables?

A: Yes, you can use a computer program to solve linear equations with two variables. However, it is always a good idea to check your solution by plugging it back into the original equation.

Q: How do I solve a linear equation with two variables when there are negative numbers involved?

A: To solve a linear equation with two variables when there are negative numbers involved, you need to isolate the variable on one side of the equation and then simplify the expression.

Q: Can I solve a linear equation with two variables using a spreadsheet?

A: Yes, you can solve a linear equation with two variables using a spreadsheet. However, it is always a good idea to check your solution by plugging it back into the original equation.

Q: How do I solve a linear equation with two variables when there are variables on both sides of the equation?

A: To solve a linear equation with two variables when there are variables on both sides of the equation, you need to isolate one variable on one side of the equation and then simplify the expression.

Q: Can I use a graphing calculator to solve a system of linear equations with two variables?

A: Yes, you can use a graphing calculator to solve a system of linear equations with two variables. However, it is always a good idea to check your solution by plugging it back into the original equations.

Q: How do I solve a linear equation with two variables when there are absolute values involved?

A: To solve a linear equation with two variables when there are absolute values involved, you need to isolate the variable on one side of the equation and then simplify the expression.

Q: Can I use a computer program to solve a system of linear equations with two variables?

A: Yes, you can use a computer program to solve a system of linear equations with two variables. However, it is always a good idea to check your solution by plugging it back into the original equations.

Q: How do I solve a linear equation with two variables when there are inequalities involved?

A: To solve a linear equation with two variables when there are inequalities involved, you need to isolate the variable on one side of the equation and then simplify the expression.

Q: Can I use a graphing calculator to solve a linear inequality with two variables?

A: Yes, you can use a graphing calculator to solve a linear inequality with two variables. However, it is always a good idea to check your solution by plugging it back into the original inequality.

Q: How do I solve a linear equation with two variables when there are complex numbers involved?

A: To solve a linear equation with two variables when there are complex numbers involved, you need to isolate the variable on one side of the equation and then simplify the expression.

Q: Can I use a computer program to solve a linear equation with two variables when there are complex numbers involved?

A: Yes, you can use a computer program to solve a linear equation with two variables when there are complex numbers involved. However, it is always a good idea to check your solution by plugging it back into the original equation.

Q: How do I solve a linear equation with two variables when there are matrices involved?

A: To solve a linear equation with two variables when there are matrices involved, you need to isolate the variable on one side of the equation and then simplify the expression.

Q: Can I use a graphing calculator to solve a linear equation with two variables when there are matrices involved?

A: Yes, you can use a graphing calculator to solve a linear equation with two variables when there are matrices involved. However, it is always a good idea to check your solution by plugging it back into the original equation.

Q: How do I solve a linear equation with two variables when there are vectors involved?

A: To solve a linear equation with two variables when there are vectors involved, you need to isolate the variable on one side of the equation and then simplify the expression.

Q: Can I use a computer program to solve a linear equation with two variables when there are vectors involved?

A: Yes, you can use a computer program to solve a linear equation with two variables when there are vectors involved. However, it is always a good idea to check your solution by plugging it back into the original equation.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving linear equations with two variables. We have covered topics such as how to solve linear equations with two variables, common mistakes to avoid, and real-world applications. We have also provided examples and explanations to help you understand the concepts. By following the steps outlined in this article, you should be able to solve linear equations with two variables and find the value of one variable.