Problem 39 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. $$ 4.21 x+5.39=12.07(2.01-4.72 x) $$

Step-by-Step Solution

Answer

The solution to the equation is approximately \(x \approx 0.31\).

Step 1: Distribute

Multiply 12.07 by each term inside the parentheses on the right side of the equation. This gives: \[4.21x + 5.39 = 12.07 \cdot 2.01 - 12.07 \cdot 4.72x\]

Step 2: Simplify

Perform the multiplication, then combine like terms:\[4.21x + 5.39 = 24.2607 - 57.0104x\] Combine like terms to form:\[61.2204x = 18.8707\]

Step 3: Solve for x

To solve for x, divide both sides of the equation by 61.2204:\[x = 18.8707 \div 61.2204\]

Step 4: Round to the nearest hundredth

Perform the division and round to the nearest hundredth to get the final result:\[x \approx 0.31\]

Key Concepts

Distribution in Algebra

Distribution is a critical algebraic technique used to simplify expressions and solve equations. Imagine you have an equation with terms inside parentheses, like in our original equation: \[4.21x + 5.39 = 12.07(2.01 - 4.72x)\] To "distribute" means to multiply each term inside the parentheses by the number or expression outside. This step is crucial because it helps to break down complex expressions into simpler forms.
In our example, we multiply 12.07 by 2.01 and 12.07 by \(-4.72x\).

• You perform \(12.07 \times 2.01\) which gives you \(24.2607\).
• Next, calculate \(12.07 \times -4.72\) to get \(-57.0104x\).

The distribution process is important because it lays the groundwork for combining like terms and simplifying the equation further.

Combining Like Terms

Combining like terms involves simplifying algebraic expressions. It means to group together terms that have the same variable raised to the same power, often making equations easier to solve. After distribution, the equation looks like this: \[4.21x + 5.39 = 24.2607 - 57.0104x\]
At this point, we want to combine like terms, which are on opposite sides of the equation.

• Notice that we have \(4.21x\) on the left and \(-57.0104x\) on the right.
• This step involves moving terms involving \(x\) to one side. Thus, add \(57.0104x\) to both sides to achieve a term of \(61.2204x\) on one side.
• Now, the equation is \(61.2204x = 18.8707\).

Combining like terms simplifies your equation even further, making it ready for the final steps of solving it.

Rounding Decimals

Rounding is the process of reducing the number of significant digits in a number while keeping its value close to what it originally was. This is often required to give an approximate answer that’s easier to communicate and understand.
In solving our equation, we've simplified down to: \[x = 18.8707 \div 61.2204\]
Following division, our result is a number with many decimal places. Here's how to round to the nearest hundredth:

• Perform the division to get \(x \approx 0.30832\).
• Look at the third decimal digit (8 in this case) to decide rounding.
• If it's 5 or higher, round the second decimal digit up. Here, round \(0.30832\) to get \(x = 0.31\).

Rounding allows us to present a neat, precise answer that aligns closely with the raw computed value.