Solve The Equation:$\[ 5\left(2+\frac{x}{5}\right)=3x \\]

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation involving fractions. The equation we will be working with is 5(2+x5)=3x5\left(2+\frac{x}{5}\right)=3x. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, xx) is 1. The equation involves fractions, which can make it more challenging to solve.

Breaking Down the Equation

Distributive Property

To solve the equation, we need to start by simplifying the left-hand side. We can do this by applying the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. In this case, we have 5(2+x5)5\left(2+\frac{x}{5}\right), which can be simplified using the distributive property.

5\left(2+\frac{x}{5}\right) = 5(2) + 5\left(\frac{x}{5}\right)

Simplifying the Left-Hand Side

Now that we have applied the distributive property, we can simplify the left-hand side of the equation.

5(2) + 5\left(\frac{x}{5}\right) = 10 + x

Solving for x

Now that we have simplified the left-hand side of the equation, we can set it equal to the right-hand side and solve for xx.

10 + x = 3x

Subtracting x from Both Sides

To isolate xx, we need to get all the terms involving xx on one side of the equation. We can do this by subtracting xx from both sides.

10 = 3x - x

Simplifying the Right-Hand Side

Now that we have subtracted xx from both sides, we can simplify the right-hand side of the equation.

10 = 2x

Dividing Both Sides by 2

Finally, we can solve for xx by dividing both sides of the equation by 2.

x = \frac{10}{2}

Conclusion

In this article, we have solved a linear equation involving fractions. We started by simplifying the left-hand side of the equation using the distributive property, and then we solved for xx by isolating it on one side of the equation. The final solution is x=5x = 5. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.

Q: How do I simplify the left-hand side of the equation?

A: To simplify the left-hand side of the equation, you can apply the distributive property and then combine like terms.

Q: How do I solve for x?

A: To solve for xx, you need to isolate it on one side of the equation. You can do this by adding or subtracting the same value from both sides, or by multiplying or dividing both sides by the same non-zero value.

Final Answer

The final answer is: 5\boxed{5}

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation involving fractions. The equation we will be working with is 5(2+x5)=3x5\left(2+\frac{x}{5}\right)=3x. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, xx) is 1. The equation involves fractions, which can make it more challenging to solve.

Breaking Down the Equation

Distributive Property

To solve the equation, we need to start by simplifying the left-hand side. We can do this by applying the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. In this case, we have 5(2+x5)5\left(2+\frac{x}{5}\right), which can be simplified using the distributive property.

5\left(2+\frac{x}{5}\right) = 5(2) + 5\left(\frac{x}{5}\right)

Simplifying the Left-Hand Side

Now that we have applied the distributive property, we can simplify the left-hand side of the equation.

5(2) + 5\left(\frac{x}{5}\right) = 10 + x

Solving for x

Now that we have simplified the left-hand side of the equation, we can set it equal to the right-hand side and solve for xx.

10 + x = 3x

Subtracting x from Both Sides

To isolate xx, we need to get all the terms involving xx on one side of the equation. We can do this by subtracting xx from both sides.

10 = 3x - x

Simplifying the Right-Hand Side

Now that we have subtracted xx from both sides, we can simplify the right-hand side of the equation.

10 = 2x

Dividing Both Sides by 2

Finally, we can solve for xx by dividing both sides of the equation by 2.

x = \frac{10}{2}

Conclusion

In this article, we have solved a linear equation involving fractions. We started by simplifying the left-hand side of the equation using the distributive property, and then we solved for xx by isolating it on one side of the equation. The final solution is x=5x = 5. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.

Q: How do I simplify the left-hand side of the equation?

A: To simplify the left-hand side of the equation, you can apply the distributive property and then combine like terms.

Q: How do I solve for x?

A: To solve for xx, you need to isolate it on one side of the equation. You can do this by adding or subtracting the same value from both sides, or by multiplying or dividing both sides by the same non-zero value.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

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

A: To determine if an equation is linear or quadratic, you need to look at the highest power of the variable. If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q: Can I use the distributive property to simplify a quadratic equation?

A: No, the distributive property can only be used to simplify linear equations. To simplify a quadratic equation, you need to use other techniques, such as factoring or completing the square.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation, while a system of linear equations is a set of two or more equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use techniques such as substitution or elimination to find the values of the variables.

Final Answer

The final answer is: 5\boxed{5}