Problem 60 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$1.6 x+0.25(12-x)=0.43(-12)$$

Step-by-Step Solution

Answer

Using the above steps, the final solution can be calculated to be \(x \approx 1.6\)

Step 1: Simplify the Equation

The equation given is \(1.6 x+0.25(12-x)=-0.43 \times -12\). First of all, process the multiplication on both sides to get a simplified equation. This provides: \(1.6x + 0.25 \times 12 - 0.25 \times x = -0.43 \times -12\).

Step 2: Further Simplifications

Now perform the multiplications to simplify the equation further: \(1.6x + 3 - 0.25x = 5.16\). Then, simplify the left-hand side by combining like terms, which gives: \((1.6-0.25)x + 3.00 = 5.16\).

Step 3: Solve for the variable

This equation translates into \(1.35x + 3.00 = 5.16\). Isolate the x term by subtracting 3 from both sides, giving \(1.35x = 5.16 - 3.00\), which simplifies to \(1.35x = 2.16\). Finally, solve for x by dividing by \(1.35\) on both sides, which provides the final solution: \(x = \frac{2.16}{1.35}\).

Key Concepts

Equation Simplification

Simplifying an equation is like cleaning up a messy room. You're getting rid of clutter to see what's important. In math, this means making the equation easier to understand by combining like terms and performing operations step by step. For the given equation, \(1.6x + 0.25(12-x) = 0.43(-12)\), we start by distributing the 0.25 and -0.43 on each side. When you distribute, you multiply each term inside the parentheses by the number outside.
For example, 0.25 times 12 gives 3, and 0.25 times \(-x\) gives \(-0.25x\). So the equation becomes \(1.6x + 3 - 0.25x = -5.16\).
This step is important because it helps us clean up the terms, making it easier to combine them in the next steps. We've configured all terms into a neater equation. Now everything is ready for further simplification.

Algebraic Manipulation

Algebraic manipulation is like rearranging pieces of a puzzle to fit perfectly. We often look for like terms to combine them. In our case, the equation is simplified to \(1.6x + 3 - 0.25x = 5.16\). Our goal is to isolate \(x\).
First, identify and combine like terms — terms with \(x\) in them. Here we combine \(1.6x\) and \(-0.25x\).
This addition gives us \(1.35x\), and our equation becomes \(1.35x + 3 = 5.16\).

• Next step: isolate \(x\) by moving other numbers to the opposite side. We subtract 3 from both sides to keep the equation balanced.
• This subtraction leads us to \(1.35x = 2.16\).

To solve for \(x\), divide both sides by 1.35. This operation gives \(x = \frac{2.16}{1.35}\).
Through careful manipulation, we uncovered the value of \(x\). It’s like solving a mystery!

Rounding Decimals

Rounding decimals is a simple yet crucial step in many problems. It's about finding a number that's easier to use, while still being close enough to the actual value. After we solve \(x = \frac{2.16}{1.35}\), we get a decimal value.
Often, you might see a long number like 1.6 or 1.722. Rounding helps us tidy this up. When asked to round to two decimal places, look at the third digit. If it's 5 or higher, round up the second digit. If it's 4 or lower, keep the second digit the same.

• In this case, after performing the division: if the result is \(1.6\), no rounding is needed as it already has one decimal place.
• If the result was \(1.605\), you'd round to \(1.61\).

Rounding makes numbers friendlier for estimation and comparisons. It's like smoothing out the edges of a rough diamond, keeping the value close yet accessible.