Problem 47 - Prealgebra

Question

Solve the equation. \(-1.7-5.56(x+6.1)=12.2\)

Step-by-Step Solution

Answer

The solution is approximately \(x = -8.601\).

Step 1: Distribute the coefficient

Multiply the coefficient -5.56 by each term inside the parentheses. The equation becomes: \(-1.7 - 5.56x - 5.56 \times 6.1 = 12.2\). Calculate \(-5.56 \times 6.1\) which equals \(-33.916\). The equation now is:\(-1.7 - 5.56x - 33.916 = 12.2\).

Step 2: Combine like terms

Combine the constants on the left side of the equation: \(-1.7 - 33.916 = -35.616\). The equation simplifies to:\(-5.56x - 35.616 = 12.2\).

Step 3: Isolate the variable term

Add 35.616 to both sides of the equation to isolate the term with x:\(-5.56x = 12.2 + 35.616\).This simplifies to:\(-5.56x = 47.816\).

Step 4: Solve for x

Divide both sides of the equation by -5.56 to solve for x:\(x = \frac{47.816}{-5.56}\).Calculate the division:\(x \approx -8.601\).

Key Concepts

Distributive Property

The distributive property is a fundamental component in algebra used to eliminate parentheses in an equation. Essentially, it states that a multiplier outside a parenthesis needs to be applied to each term within the parenthesis. This is often stated as:

• \( a(b + c) = ab + ac \)

Using the distributive property helps convert complex equations into simpler forms that can be more easily manipulated. In our exercise, we have \(-5.56(x + 6.1)\), where \(-5.56\) is the coefficient that needs to be distributed across both \(x\) and \(6.1\).
This results in two separate terms: \(-5.56x\) and \(-5.56 \times 6.1\). By performing the multiplication, we simplify the expression significantly. For instance, calculating \(-5.56 \times 6.1\) gives \(-33.916\), transforming our equation step by step.

Solving Equations

Solving equations involves a series of steps to find the value of the unknown variable, often represented as \(x\). The goal is to simplify the equation until the variable is isolated and its value can be determined. A well-structured approach can demystify this process:

• Start by distributing any coefficients to eliminate parentheses.
• Combine like terms to further simplify.

In the given exercise, we first used the distributive property, then moved on to combining like terms: \(-1.7\) and \(-33.916\), resulting in \(-35.616\).
This simplification is essential as it reduces the equation complexity, bringing us closer to isolating the variable.

Isolation of Variables

The ultimate aim when solving equations is the isolation of the variable. This means manipulating the equation until the variable appears by itself on one side of the equation.
In the provided equation, after simplifying using the distributive property and combining like terms, we had \(-5.56x - 35.616 = 12.2\). To isolate \(x\), we add \(35.616\) to both sides.

• This action cancels out \(-35.616\) on the left side, leaving \(-5.56x\) on its own.
• The right side then becomes \(12.2 + 35.616\), which equals \(47.816\).

Finally, to completely isolate \(x\), divide both sides by \(-5.56\). Through this division, the calculation \(x = \frac{47.816}{-5.56}\) yields \(x \approx -8.601\). Thus, this process unveils the unknown's value, highlighting how each step contributes to the solution.