Problem 52 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$16-2.4 x=-8$$

Step-by-Step Solution

Answer

Applying the steps outlined above results in \(x = 10.00\).

Step 1: Isolate the Term Involving x.

To isolate \(x\), begin by moving the term that doesn't involve \(x\) to the opposite side of the equality. This done by adding 2.4x to both sides of the equation, giving \[-2.4x + 16 = -8 \rightarrow -2.4x = -8 - 16.\]

Step 2: Simplify the Right-hand Side.

Perform the subtraction on the right-hand side to get a number: \[-2.4x = -8 - 16 \rightarrow -2.4x = -24.\]

Step 3: Solve for x.

To find the value of \(x\), divide both sides of the equation by -2.4 to isolate \(x\): \[-2.4x / -2.4 = -24 / -2.4 \rightarrow x = 10.\]

Step 4: Round the Solution.

Round your answer to two decimal places, but in this particular case, \(x = 10.00\).

Key Concepts

Algebraic Operations

Understanding algebraic operations is crucial for solving linear equations efficiently. At the heart of these operations are the basic mathematical processes: addition, subtraction, multiplication, and division. In the context of solving equations, these operations are used to manipulate the equation in order to isolate the unknown variable.

For instance, in the given exercise \( 16-2.4x=-8 \) when we talk about moving or 'transposing' terms, we mean using these operations to transfer a term from one side of the equation to the other to help isolate the variable. This is done in a way that maintains the equality of the equation. If you add or subtract a number from one side, you must do the same to the other side. Similarly, if you multiply or divide one side by a number, you repeat that process on the opposite side. These are indispensable tools for solving equations and a strong grasp on them will make algebra much more approachable.

Isolating Variables

The goal of isolating variables in an equation is to find the value of the unknown variable. This process involves using algebraic operations to get the variable by itself on one side of the equation. A step-by-step method often leads to success.

In the equation \( 16-2.4x=-8 \) from the exercise, we start by eliminating the constants from the side with the variable. In this case, we add \( 2.4x \) to both sides to move the constant number to the other side—practically 'freeing' the variable. Then, we're left with \( -2.4x = -24 \) Simplifying the equation further by dividing both sides by \( -2.4 \) allows us to reach the variable \( x \) standing alone, leading to the answer. Ly breaking down the process into manageable steps, we prevent errors and ensure a clear path to the solution.

Rounding Decimals

In many real-world situations, and also in mathematics problems, there is a need to round decimals to a certain precision. Rounding decimals is the process of adjusting the number to reduce the length after the decimal point according to specified rules, typically to make the number simpler to work with.

The rules are straightforward: if the digit right after the cutoff point (where you want to round) is 5 or higher, you increase the last retained digit by one. If it's lower than 5, you leave the last retained digit as it is. In our exercise, though, rounding \( x = 10.00 \) to two decimal places does not alter the number since the hundredths and thousandths places are zeros. This is a unique case, as usually, rounding will adjust the number. Always pay attention to the rounding instruction as it ensures the precision of your answer aligns with the question's requirement.