Problem 48 - Prealgebra

Question

Solve the equation. \(-7.93+0.01(x+7.9)=14.2\)

Step-by-Step Solution

Answer

The solution is \(x = 2205.1\).

Step 1: Expand the Expression

Start by expanding the equation \(-7.93 + 0.01(x + 7.9) = 14.2\). Distribute the \(0.01\) across the terms inside the parenthesis:\(-7.93 + 0.01x + 0.079 = 14.2\).

Step 2: Combine Like Terms

Combine the constants on the left side of the equation:\(-7.93 + 0.079 = -7.851\).The equation now reads:\(-7.851 + 0.01x = 14.2\).

Step 3: Isolate the Variable Term

Add \(7.851\) to both sides to isolate the term with \(x\):\(0.01x = 14.2 + 7.851\).This simplifies to:\(0.01x = 22.051\).

Step 4: Solve for x

To isolate \(x\), divide both sides by \(0.01\):\(x = \frac{22.051}{0.01}\).This simplifies to:\(x = 2205.1\).

Key Concepts

Expanding Expressions

When solving linear equations, expanding expressions is often the first step. In the equation \(-7.93 + 0.01(x + 7.9) = 14.2\), you need to distribute the term \(0.01\) to each part inside the parentheses. This means multiplying both terms, \(x\) and \(7.9\), by \(0.01\).
This process effectively eliminates the parentheses and simplifies the equation, setting the stage for solving it.

• The term \(0.01x\) is the result of multiplying \(0.01\) by \(x\).
• The term \(0.079\) is the result of multiplying \(0.01\) by \(7.9\).

After distributing, the expanded expression becomes \(-7.93 + 0.01x + 0.079 = 14.2\).
Expanding expressions helps to break down complex parts of an equation into simpler, more manageable terms.

Combining Like Terms

After expanding an expression in a linear equation, you often encounter like terms that can be combined. In our case, like terms on the left-hand side of the equation are the constant terms: \(-7.93\) and \(0.079\).
Combining these like terms simplifies the equation further.

To combine the constants:

• Simply add or subtract them depending on their signs.
• In the example, we have \(-7.93 + 0.079 = -7.851\).

This leaves us with the simplified equation: \(-7.851 + 0.01x = 14.2\).
Combining like terms reduces the number of terms in an equation, making it easier to find the solution.

Isolating Variables

Once the equation is simplified, the next key step is to isolate the variable, which means getting the variable \(x\) on one side of the equation on its own.
This is crucial for finding the value of \(x\). In our equation, \(-7.851 + 0.01x = 14.2\), isolating the variable involves moving all other terms to the right side.

• First, add \(7.851\) to both sides to cancel it out on the left side: \(0.01x = 14.2 + 7.851\).
• Now, the equation simplifies to \(0.01x = 22.051\).

Finally, to completely isolate \(x\), divide both sides by \(0.01\): \(x = \frac{22.051}{0.01}\).
This gives \(x = 2205.1\).
Isolating the variable is the step where we solve the equation and find the value of \(x\), leading us to the solution.