Problem 80 - Algebra and Trigonometry

Question

Find the solution of the equation rounded to two decimals. \(2.14(x-4.06)=2.27-0.11 x\)

Step-by-Step Solution

Answer

The solution to the equation is approximately 4.86.

Step 1: Expand the Equation

First, expand the left side of the equation using the distributive property. Multiply 2.14 by both terms inside the parentheses: \[ 2.14(x - 4.06) = 2.14x - 8.6684 \] Now the equation becomes: \[ 2.14x - 8.6684 = 2.27 - 0.11x \]

Step 2: Combine Like Terms

Next, move the term with \(x\) on the right side of the equation to the left side by adding \(0.11x\) to both sides. This results in: \[ 2.14x + 0.11x - 8.6684 = 2.27 \] Combine the \(x\) terms: \[ 2.25x - 8.6684 = 2.27 \]

Step 3: Isolate the x Term

To isolate the term with \(x\), add 8.6684 to both sides of the equation: \[ 2.25x = 2.27 + 8.6684 \] Calculate the right side: \[ 2.25x = 10.9384 \]

Step 4: Solve for x

Finally, divide both sides of the equation by 2.25 to solve for \(x\): \[ x = \frac{10.9384}{2.25} \] Calculate the result and round it to two decimal places: \[ x \approx 4.86 \]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows us to multiply a single term across terms inside a bracket. By applying this property, expressions can be simplified, making complex equations easier to work with. In the context of the equation \(2.14(x-4.06) = 2.27 - 0.11x\), we used the distributive property to expand \(2.14(x-4.06)\). Here’s how it works:

• Multiply 2.14 by \(x\).
• Multiply 2.14 by -4.06.

This means the equation becomes \(2.14x - 8.6684\). This step sets the stage for later simplifications in solving the equation. By distributing, we break down complex expressions into manageable parts.

Combining Like Terms

After using the distributive property, the next step is to combine like terms. This means simplifying the equation further by adding or subtracting terms that are similar.
In our equation, after expanding, we have \(2.14x - 8.6684 = 2.27 - 0.11x\). To simplify, we move terms containing \(x\) to one side. We do this by adding \(0.11x\) to both sides:

• Adding \(0.11x\) to \(2.14x\) results in \(2.25x\).
• The constant term \(-8.6684\) stays with \(x\) terms.

Now the equation is \(2.25x - 8.6684 = 2.27\). Combining like terms helps transform the equation into a simpler form, facilitating the solving process.

Isolating Variables

Isolating the variable is a key step in solving equations. It involves manipulating the equation to get the variable alone on one side. This step allows us to determine its value. Once like terms are combined and our equation is \(2.25x - 8.6684 = 2.27\), we need to isolate \(x\).
To do that, simply add 8.6684 to both sides of the equation:

• This moves all constants to the right side.
• We're left with \(2.25x = 10.9384\).

This step ensures that \(x\) is separated and ready for solving. Isolating the variable is foundational in finding the solution to any algebraic equation.

Rounding Numbers

Rounding numbers is often employed to communicate solutions with precision and clarity. In real-world applications, answers are frequently rounded to a set number of decimal places for consistency.
Once you’ve isolated the variable and performed final computations, as in \(x = \frac{10.9384}{2.25}\), the last step is to round \(x\) to two decimal places:

• First, compute \(x\) which is approximately 4.86026667.
• Look at the third decimal place to decide rounding.
• Since it's below 5, \(x\) rounds to 4.86.

Rounding numbers helps in clearly maintaining a desired level of accuracy and simplifies the presentation of final answers.