Problem 2 - Technical Mathematics with Calculus

Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$3 x-2=10$$

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Add 2 to both sides

To isolate the term containing the variable, add 2 to both sides of the equation. This will give us: $$3x - 2 + 2 = 10 + 2$$ which simplifies to: $$3x = 12$$

Step 2: Divide both sides by 3

Now that we have a single term with the variable on one side, we divide both sides by 3 to isolate x: $$\frac{3x}{3} = \frac{12}{3}$$ This simplifies to: $$x = 4$$

Step 3: Check the solution

To verify if x = 4 is a solution, substitute x with 4 in the original equation and check if both sides are equal: $$3(4) - 2 = 10$$ which simplifies to: $$12 - 2 = 10$$ $$10 = 10$$ Since both sides are equal, x = 4 is a verified solution.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They consist of variables, numbers, and arithmetic operations. The puzzle-like nature of algebraic equations demands that we find the value(s) of the variable(s) that make the equation true.

For example, consider the equation \(3x - 2 = 10\). Here, \(3x - 2\) and \(10\) are two expressions separated by an equals sign, indicating they are equal when the variable \(x\) takes on a certain value. Solving an algebraic equation is like unraveling a mystery where the culprit is the unknown variable hidden within the numerical details.

We aim to find this 'culprit' or solution by performing a series of operations that simplify the equation and slowly bring the variable out into the light. A key strategy involves performing the same operations on both sides to maintain the balance, since equality is like a scale in perfect balance, any change on one side must be matched on the other.

Isolating Variables

Isolating the variable in an algebraic equation is crucial to solving for the unknown. The goal is to manipulate the equation to have the variable on one side and all other numerical values on the other. Doing so makes it explicitly clear what the value of the variable is.

Let's dissect the process with the given equation \(3x - 2 = 10\). The first step often involves undoing any addition or subtraction to free the term with the variable. By adding 2 to both sides \(3x - 2 + 2 = 10 + 2\), we simplify the equation to \(3x = 12\).

Dividing to Uncover the Variable

Next, any multiplication or division is undone to isolate the variable completely. In dividing both sides by 3 \(\frac{3x}{3} = \frac{12}{3}\), the variable \(x\) stands alone, and we are left with \(x = 4\).

Properly isolating the variable is like finding the key to a locked door; once turned, everything falls into place, revealing the solution to our algebraic dilemma.

Equation Checking

Whenever we find a solution to an algebraic equation, it's essential to make sure the solution is correct. This process of verification is known as equation checking.

To check the solution of the equation \(3x - 2 = 10\) with the suspected solution \(x = 4\), we substitute the value of the variable back into the original equation. It is a litmus test for our solution and works by observing if the equation holds true: \(3(4) - 2 = 10\) simplifies to form \(12 - 2 = 10\), which further simplifies to \(10 = 10\). Since both sides match, we have confirmed that our solution, \(x = 4\), satisfies the original equation.

The Importance of Checking

Checking not only verifies the accuracy of our solution but also ensures that no arithmetic errors were made during the problem-solving process. It's akin to a chef tasting their dish before serving; it's the final step that assures the desired result has been achieved.