Problem 42 - Prealgebra

Question

Solve the equation. \(-0.7(0.2 x+5.48)=16.45\)

Step-by-Step Solution

Answer

The solution to the equation is approximately \( x \approx -144.9 \).

Step 1: Removing Parentheses

We'll start by distributing \(-0.7\) into the expression inside the parentheses. The expression becomes:\[ -0.7 imes 0.2x - 0.7 imes 5.48 = 16.45 \]Calculating these products, we have:\[ -0.14x - 3.836 = 16.45 \]

Step 2: Isolating the Variable Term

Next, we need to isolate the term with the variable. We will add \(3.836\) to both sides of the equation:\[ -0.14x = 16.45 + 3.836 \]Simplifying the right side gives us:\[ -0.14x = 20.286 \]

Step 3: Solving for x

To solve for \(x\), we divide both sides of the equation by \(-0.14\):\[ x = \frac{20.286}{-0.14} \]Carrying out the division, we find:\[ x \approx -144.9 \]

Key Concepts

Understanding the Distributive Property

The distributive property is a valuable tool in algebra that simplifies expressions and equations. This rule states that a number multiplied by a sum is the same as multiplying the number by each addend and then adding the products. In formula terms, it's represented as:

• \( a(b + c) = ab + ac \)

Consider the equation:

• \(-0.7(0.2x + 5.48) = 16.45\)

To apply the distributive property, multiply \(-0.7\) by both \(0.2x\) and \(5.48\):

• \(-0.7 \cdot 0.2x = -0.14x\)
• \(-0.7 \cdot 5.48 = -3.836\)

These steps simplify the equation to:

• \(-0.14x - 3.836 = 16.45\)

By understanding these steps, you can simplify and solve equations more easily.

Isolating Variable in Equations

Isolating the variable is an essential step to solving linear equations. This means rearranging the equation so that the variable is by itself on one side. Let's consider our transformed equation:

• \(-0.14x - 3.836 = 16.45\)

To isolate \(x\), we focus on eliminating the number \(-3.836\) from the left side:

• Add \(3.836\) to both sides to cancel it out:\(-0.14x = 16.45 + 3.836\)
• Calculate the sum on the right:\(-0.14x = 20.286\)

Now, \(x\) is almost alone. You've successfully removed the number besides the variable, preparing for the final step of solving.

The Role of Division in Solving Equations

Once you've isolated the variable term, it's time to solve for the variable using division. In our example, we have:

• \(-0.14x = 20.286\)

To solve for \(x\), divide both sides of the equation by the coefficient of \(x\), which is \(-0.14\):

• \(x = \frac{20.286}{-0.14}\)
• Perform the division to find:\(x \approx -144.9\)

By dividing, you scale down the coefficient next to the variable, finally isolating the variable fully. This step provides the value that satisfies the original equation. Understanding division in equations ensures you can effectively solve for variables any time they appear in linear equations.