Solve This Equation: $\frac{3}{5}(x-10)=18-4x-1$Step 1: Simplify Using The Distributive Property. Which Number Can Be Distributed Across The Two Terms Inside The Parentheses?

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation using the distributive property. The equation we will be solving is: $\frac{3}{5}(x-10)=18-4x-1$. We will break down the solution into manageable steps, making it easy for readers to follow along.

Step 1: Simplify using the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single value to multiple values inside parentheses. In this case, we need to distribute the $\frac{3}{5}$ across the two terms inside the parentheses: $(x-10)$. To do this, we need to multiply $\frac{3}{5}$ by both $x$ and $-10$.

\frac{3}{5}(x-10) = \frac{3}{5}x - \frac{3}{5}(10)

Now, we can simplify the expression by multiplying $\frac{3}{5}$ by $10$. To do this, we need to multiply the numerator and denominator by $10$.

\frac{3}{5}(10) = \frac{3 \times 10}{5} = \frac{30}{5} = 6

So, the simplified expression is:

\frac{3}{5}(x-10) = \frac{3}{5}x - 6

Step 2: Simplify the Right-Hand Side of the Equation

Now that we have simplified the left-hand side of the equation, we need to simplify the right-hand side. The right-hand side of the equation is: $18-4x-1$. We can simplify this expression by combining like terms.

18-4x-1 = 17-4x

So, the simplified equation is:

\frac{3}{5}x - 6 = 17-4x

Step 3: Add 6 to Both Sides of the Equation

To get rid of the negative term on the left-hand side of the equation, we need to add 6 to both sides of the equation.

\frac{3}{5}x - 6 + 6 = 17-4x + 6

Simplifying the left-hand side of the equation, we get:

\frac{3}{5}x = 23-4x

Step 4: Add 4x to Both Sides of the Equation

To get all the terms with $x$ on one side of the equation, we need to add $4x$ to both sides of the equation.

\frac{3}{5}x + 4x = 23-4x + 4x

Simplifying the left-hand side of the equation, we get:

\frac{3}{5}x + \frac{4}{1}x = 23

Step 5: Combine Like Terms

To combine like terms, we need to find a common denominator for the two terms on the left-hand side of the equation.

\frac{3}{5}x + \frac{20}{5}x = 23

Simplifying the left-hand side of the equation, we get:

\frac{23}{5}x = 23

Step 6: Multiply Both Sides of the Equation by 5

To get rid of the fraction on the left-hand side of the equation, we need to multiply both sides of the equation by 5.

\frac{23}{5}x \times 5 = 23 \times 5

Simplifying the left-hand side of the equation, we get:

23x = 115

Step 7: Divide Both Sides of the Equation by 23

To solve for $x$, we need to divide both sides of the equation by 23.

\frac{23x}{23} = \frac{115}{23}

Simplifying the left-hand side of the equation, we get:

x = 5

Conclusion

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Introduction

In our previous article, we solved a linear equation using the distributive property. In this article, we will provide a Q&A guide to help readers understand the solution and answer any questions they may have.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a single value to multiple values inside parentheses. It states that for any numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

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

A: To apply the distributive property to solve a linear equation, you need to follow these steps:

1. Distribute the coefficient to the terms inside the parentheses.
2. Simplify the expression by combining like terms.
3. Isolate the variable by adding or subtracting the same value to both sides of the equation.
4. Solve for the variable by dividing both sides of the equation by the coefficient.

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. For example, $2x + 3 = 5$ is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows you to solve a quadratic equation. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I choose between the distributive property and the quadratic formula?

A: To choose between the distributive property and the quadratic formula, you need to look at the type of equation you are solving. If the equation is linear, you can use the distributive property. If the equation is quadratic, you can use the quadratic formula.

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

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

• Not distributing the coefficient to the terms inside the parentheses.
• Not simplifying the expression by combining like terms.
• Not isolating the variable by adding or subtracting the same value to both sides of the equation.
• Not solving for the variable by dividing both sides of the equation by the coefficient.

Conclusion

In this article, we provided a Q&A guide to help readers understand the solution to a linear equation using the distributive property. We also discussed the quadratic formula and how to choose between the distributive property and the quadratic formula. We hope this article has been helpful in answering any questions you may have had about solving linear equations.

Additional Resources

• Khan Academy: Solving Linear Equations
• Mathway: Solving Linear Equations
• Wolfram Alpha: Solving Linear Equations

Final Thoughts

Solving linear equations is an important skill to master in mathematics. By following the steps outlined in this article, you can solve linear equations with ease. Remember to always distribute the coefficient to the terms inside the parentheses, simplify the expression by combining like terms, and isolate the variable by adding or subtracting the same value to both sides of the equation. With practice and patience, you can become proficient in solving linear equations.