Problem 58 - Prealgebra

Question

Solve the equation. \(5.5(6.7 x+7.3)=-5.5(-4.2 x+2.2)\)

Step-by-Step Solution

Answer

The solution is \(x = -3.8\).

Step 1: Distribute the Constants

Begin by applying the distributive property to both sides of the equation. For the left side, distribute the 5.5 into \((6.7x + 7.3)\). For the right side, distribute \(-5.5\) into \((-4.2x + 2.2)\). This gives us:\[5.5 \times 6.7x + 5.5 \times 7.3 = -5.5 \times -4.2x + -5.5 \times 2.2\]This simplifies to:\[36.85x + 40.15 = 23.1x - 12.1\]

Step 2: Move the Variable Terms to One Side

Next, move all terms involving \(x\) to one side of the equation by subtracting \(23.1x\) from both sides:\[36.85x - 23.1x + 40.15 = -12.1\]which simplifies to:\[13.75x + 40.15 = -12.1\]

Step 3: Isolate the Variable Term

To isolate the term with the variable \(x\), subtract 40.15 from both sides of the equation:\[13.75x = -12.1 - 40.15\]Simplifying gives us:\[13.75x = -52.25\]

Step 4: Solve for the Variable

Finally, solve for \(x\) by dividing both sides by 13.75:\[x = \frac{-52.25}{13.75}\]This simplifies to:\[x = -3.8\]

Key Concepts

Distributive Property

The distributive property is one of the fundamental algebraic principles you'll frequently encounter in mathematics. It helps simplify expressions and solve equations by distributing a multiplication across terms inside parentheses. In the equation \[5.5(6.7x + 7.3) = -5.5(-4.2x + 2.2)\], we need to apply this property to each term within the parentheses before simplifying further.

• On the left side: multiply 5.5 with both 6.7x and 7.3, which results in two terms: \(36.85x\) and \(40.15\).
• On the right side: remember to distribute -5.5 across both -4.2x and 2.2. This yields \(23.1x\) and \(-12.1\).

By using the distributive property, we transform the original equation into a simpler form: \[36.85x + 40.15 = 23.1x - 12.1\].
This method is crucial because it sets the stage for making the equation easier to handle in subsequent steps.

Isolating Variables

Isolating the variable is the process of manipulating an equation in such a way that the variable we are solving for is on one side, and all constant terms are on the other. The goal here is to have the variable, in this case, \(x\), standing alone.
In our equation \[36.85x + 40.15 = 23.1x - 12.1\], we start by bringing all terms containing \(x\) to one side by subtracting \(23.1x\) from both sides:

• This results in \(13.75x + 40.15 = -12.1\).

Once the variable terms are consolidated on one side, we then focus on removing any constants attached to the variable. In this instance, we subtract 40.15 from both sides:

• The equation becomes \(13.75x = -52.25\).

This crucial step of isolating the variable allows us to focus directly on finding the solution for \(x\) in the simplest form possible.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. After isolating the variable, the objective is to determine its precise value.
Continuing from the previous steps, our simplified equation is \[13.75x = -52.25\].
To solve for \(x\), we divide both sides by 13.75:

• This calculation yields \(x = \frac{-52.25}{13.75}\).
• Upon further simplification, the value of \(x\) is found to be \(-3.8\).

This straightforward method of solving linear equations through isolation and division ensures that you arrive at a precise solution efficiently. Understanding these steps will empower you to handle similar algebraic equations with confidence and ease.