Solve The Equation:$\[ 2x + 50 = 10y \\]

Feb 28, 2025 by ADMIN 41 views

**Solving Linear Equations: A Comprehensive Guide**

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 focus on solving a specific type of linear equation, which is given by the equation $2x + 50 = 10y$ . We will break down the solution process into manageable steps, and provide a clear explanation of each step.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is in the form of $ax + by = c$ , where $a$ , $b$ , and $c$ are constants. In this case, $a = 2$ , $b = -10$ , and $c = 50$ . The variable $x$ is the unknown quantity that we want to solve for.

Step 1: Isolate the Variable

The first step in solving the equation is to isolate the variable $x$ . To do this, we need to get rid of the constant term $50$ on the left-hand side of the equation. We can do this by subtracting $50$ from both sides of the equation.

${ 2x + 50 - 50 = 10y - 50 }$

This simplifies to:

${ 2x = 10y - 50 }$

Step 2: Get Rid of the Coefficient

The next step is to get rid of the coefficient $2$ that is multiplying the variable $x$ . We can do this by dividing both sides of the equation by $2$ .

${ \frac{2x}{2} = \frac{10y - 50}{2} }$

This simplifies to:

${ x = \frac{10y - 50}{2} }$

Step 3: Simplify the Expression

The final step is to simplify the expression on the right-hand side of the equation. We can do this by dividing the numerator and denominator by their greatest common divisor, which is $2$ .

${ x = \frac{10y - 50}{2} = \frac{5(2y - 10)}{2} = \frac{5(2y - 10)}{2} }$

This simplifies to:

${ x = 5y - 25 }$

Conclusion

In this article, we have solved the linear equation $2x + 50 = 10y$ using a step-by-step approach. We have isolated the variable $x$ , gotten rid of the coefficient, and simplified the expression. The final solution is $x = 5y - 25$ . This equation can be used to solve for the value of $x$ in terms of the variable $y$ .

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.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

Use the distributive property: The distributive property states that $a(b + c) = ab + ac$ . This can be used to simplify expressions and solve linear equations.
Use inverse operations: Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division.
Check your work: It's always a good idea to check your work by plugging the solution back into the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

Forgetting to isolate the variable: Make sure to isolate the variable on one side of the equation.
Forgetting to get rid of the coefficient: Make sure to get rid of the coefficient by dividing both sides of the equation by the coefficient.
Forgetting to simplify the expression: Make sure to simplify the expression on the right-hand side of the equation.

Conclusion

Q&A: Frequently Asked Questions

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 of $ax + by = c$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. This can be done by using inverse operations, such as addition and subtraction, and multiplication and division.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that $a(b + c) = ab + ac$ . This can be used to simplify expressions and solve linear equations.

Q: What is an inverse operation?

A: An inverse operation is an operation that "undoes" another operation. For example, addition and subtraction are inverse operations, as are multiplication and division.

Q: How do I check my work?

A: To check your work, plug the solution back into the original equation and see if it is true. If it is true, then you have solved the equation correctly.

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

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

Forgetting to isolate the variable
Forgetting to get rid of the coefficient
Forgetting to simplify the expression

Q: How do I use the distributive property to solve a linear equation?

A: To use the distributive property to solve a linear equation, multiply the coefficient of the variable by the term inside the parentheses, and then add or subtract the result from the other side of the equation.

Q: How do I use inverse operations to solve a linear equation?

A: To use inverse operations to solve a linear equation, use the inverse operation of the operation that is being performed on the variable. For example, if the equation is $x + 3 = 5$ , you can use the inverse operation of addition, which is subtraction, to solve for $x$ .

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.

Q: How do I simplify an expression?

A: To simplify an expression, combine like terms and eliminate any unnecessary parentheses.

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 solve a system of linear equations?

A: To solve a system of linear equations, use the method of substitution or elimination to find the values of the variables.

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step approach outlined in this article, you can solve linear equations with ease. Remember to isolate the variable, get rid of the coefficient, and simplify the expression. With practice and patience, you will become proficient in solving linear equations and be able to apply them to real-world problems.