Consider The Equation $0.2(x-4.5)+1.7=9.6$.1. Use The Distributive Property.2. Combine Like Terms.3. Use The Properties Of Equality And Inverse Operations To Isolate The Variable.$ \begin{array}{l} 0.2x + (-0.9) + 1.7 = 9.6 \\ 0.2x + 0.8

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, combining like terms, and using the properties of equality and inverse operations to isolate the variable.

The Equation

The given equation is:

$$0.2(x-4.5)+1.7=9.6$$

Our goal is to isolate the variable $x$ and find its value.

Step 1: Use the Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, we have:

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

We can apply this property to the given equation by distributing the coefficient $0.2$ to the terms inside the parentheses:

$$0.2(x-4.5)+1.7=9.6$$

$$0.2x - 0.9 + 1.7 = 9.6$$

Step 2: Combine Like Terms

Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: $-0.9$ and $1.7$. We can combine them by adding their coefficients:

$$0.2x - 0.9 + 1.7 = 9.6$$

$$0.2x + 0.8 = 9.6$$

Step 3: Use the Properties of Equality and Inverse Operations to Isolate the Variable

The properties of equality state that if two expressions are equal, then we can perform the same operations on both sides of the equation without changing the equality. In this case, we want to isolate the variable $x$, so we will use inverse operations to get rid of the coefficient $0.2$.

To do this, we can multiply both sides of the equation by the reciprocal of $0.2$, which is $\frac{1}{0.2} = 5$. This will cancel out the coefficient $0.2$ and leave us with the variable $x$:

$$0.2x + 0.8 = 9.6$$

$$5(0.2x + 0.8) = 5(9.6)$$

$$x + 4 = 48$$

Solving for $x$

Now that we have isolated the variable $x$, we can solve for its value by subtracting $4$ from both sides of the equation:

$$x + 4 = 48$$

$$x = 48 - 4$$

$$x = 44$$

Therefore, the value of the variable $x$ is $44$.

Conclusion

Introduction

In our previous article, we walked through the steps to solve a linear equation using the distributive property, combining like terms, and applying the properties of equality and inverse operations to isolate the variable. In this article, we will answer some common questions that students may have when solving linear equations.

Q: What is the distributive property, and how is it used in solving linear equations?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, we have:

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

In the context of solving linear equations, the distributive property is used to expand expressions inside parentheses. For example, in the equation:

$$0.2(x-4.5)+1.7=9.6$$

We can apply the distributive property to expand the expression inside the parentheses:

$$0.2x - 0.9 + 1.7 = 9.6$$

Q: What are like terms, and how are they combined in solving linear equations?

A: Like terms are terms that have the same variable raised to the same power. In the equation:

$$0.2x - 0.9 + 1.7 = 9.6$$

We have two like terms: $-0.9$ and $1.7$. We can combine them by adding their coefficients:

$$0.2x + 0.8 = 9.6$$

Q: What are the properties of equality, and how are they used in solving linear equations?

A: The properties of equality state that if two expressions are equal, then we can perform the same operations on both sides of the equation without changing the equality. In the equation:

$$0.2x + 0.8 = 9.6$$

We can multiply both sides of the equation by the reciprocal of $0.2$, which is $\frac{1}{0.2} = 5$. This will cancel out the coefficient $0.2$ and leave us with the variable $x$:

$$5(0.2x + 0.8) = 5(9.6)$$

Q: How do I know which operation to perform first when solving a linear equation?

A: When solving a linear equation, it's essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following this order of operations, you can ensure that you're solving the equation correctly.

Q: What if I have a fraction as a coefficient in the equation? How do I handle it?

A: If you have a fraction as a coefficient in the equation, you can multiply both sides of the equation by the reciprocal of the fraction to eliminate it. For example, if you have the equation:

$$\frac{1}{2}x + 3 = 5$$

You can multiply both sides of the equation by $2$ to eliminate the fraction:

$$2(\frac{1}{2}x + 3) = 2(5)$$

Conclusion

Solving linear equations requires a step-by-step approach, using the distributive property, combining like terms, and applying the properties of equality and inverse operations to isolate the variable. By following these steps and understanding the properties of equality, you can solve equations like the one given in this article and find the value of the variable.