Problem 38 - Prealgebra

Question

Solve the equation. \(0.5(-2.5 x-4.7)=16.9\)

Step-by-Step Solution

Answer

The solution is \(x = -15.4\).

Step 1: Expand the Equation

First, let's remove the parenthesis by distributing the 0.5 to each term inside the parenthesis. This means we multiply 0.5 with each term:\[0.5(-2.5x) + 0.5(-4.7)\]This results in:\[-1.25x - 2.35\]

Step 2: Set the Equation to 0

Now, equate the expanded expression to 16.9:\[-1.25x - 2.35 = 16.9\]

Step 3: Isolate the x-term

Add 2.35 to both sides to get the equation with the x-term isolated:\[-1.25x = 16.9 + 2.35\]Calculate the right side:\[-1.25x = 19.25\]

Step 4: Solve for x

Divide both sides by -1.25 to solve for x:\[x = \frac{19.25}{-1.25}\]Calculate the division:\[x = -15.4\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term with each term inside a parenthesis. In this exercise, we have the expression \(0.5(-2.5x - 4.7)\). The distributive property lets us "distribute" the \(0.5\) to both \(-2.5x\) and \(-4.7\).

• First, multiply \(0.5\) by \(-2.5x\). This results in \(-1.25x\).
• Next, multiply \(0.5\) by \(-4.7\). This gives \(-2.35\).

Putting it all together, the expression \(0.5(-2.5x - 4.7)\) becomes \(-1.25x - 2.35\).Applying the distributive property helps us simplify equations, making it easier to move to the next steps in solving.

Isolating Variables

One of the primary goals in solving equations is to isolate the variable, making it possible to solve for its value. Once the equation \(-1.25x - 2.35 = 16.9\) is simplified using the distributive property, it's time to focus on getting the \(x\)-term by itself.

• To start, add \(2.35\) to both sides of the equation. This helps move the constant term to the other side, resulting in \(-1.25x = 16.9 + 2.35\).
• Perform the addition on the right side of the equation, yielding \(-1.25x = 19.25\).

By isolating \(-1.25x\), we are one step closer to finding the value of \(x\). This step is crucial in solving linear equations, as it simplifies further calculations.

Division of Decimals

Dividing decimals can seem daunting, but with clear steps, it can be straightforward. When we reached the equation \(-1.25x = 19.25\), the final step was to solve for \(x\).

• Divide both sides of the equation by \(-1.25\) to isolate \(x\). This gives \(x = \frac{19.25}{-1.25}\).
• When dividing decimals, it's useful to move the decimal point to the right to make the division easier, aligning with standard division practices.

This division yields \(x = -15.4\). Mastering decimal operations is vital for tackling equations like these efficiently.