Problem 9 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ 3 x=42 $$

Step-by-Step Solution

Answer

Answer: The value of x in the given conditional equation is 14.

Step 1: Identify the Equation and Coefficient of x

Identify the given equation and the coefficient of x. The given equation is: $$ 3x = 42 $$ The coefficient of x is 3.

Step 2: Isolate x

To isolate x, divide both sides of the equation by the coefficient of x, which is 3. $$ \frac{3x}{3} = \frac{42}{3} $$

Step 3: Simplify

Simplify the equation to find the value of x: $$ x = 14 $$ The solution of the conditional equation is x = 14.

Key Concepts

Conditional Equations

Conditional equations are types of equations that hold true only for particular values of the variable. These equations are not universally true, meaning that they do not apply under all circumstances or for all values. For example, in the equation \(3x = 42\), the equation is satisfied only when \(x = 14\).

• Example: Take the equation \(x + 5 = 10\). The solution \(x = 5\) is the only value that satisfies this conditional equation.
• Understanding that these equations are conditional helps in recognizing that a solution can indeed be identified, unlike contradictions or identities which never or always hold.

To solve such conditional equations, one should manipulate the equation to determine the specific values for which it holds true.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to solve for unknown variables. It involves using basic arithmetic operations such as addition, subtraction, multiplication, and division.

• Example: For \(3x = 42\), algebraic manipulation tells us to divide both sides by 3. This simplifies the equation to \(x = 14\).
• Ensuring all operations done on one side are also done on the other maintains the balance of the equation.

This process is crucial in solving equations as it allows us to transform complex expressions into simple expressions that make the solution apparent.
It's important to perform these operations carefully to avoid errors which could lead to incorrect solutions.

Isolation of Variables

Isolation of variables is a fundamental step in solving equations. It refers to the process of getting the variable by itself on one side of the equation. By doing this, the equation becomes easy to solve as it clearly shows the solution.

• Example: In \(3x = 42\), dividing both sides by 3 isolates \(x\) by performing \( \frac{3x}{3} = \frac{42}{3} \).
• The ultimate goal is to reach an equation that reads \(x = \text{some number}\).

Successfully isolating variables is crucial as it clarifies the exact value of the unknown variable that satisfies the equation.
This method is widely used because it provides a straightforward way of finding solutions.