Problem 84 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 3 x+2=7 $$

Step-by-Step Solution

Answer

\( x = \frac{5}{3} \)

Step 1: Isolate the variable term

To isolate the variable term, subtract 2 from both sides of the equation: \[ 3x + 2 - 2 = 7 - 2 \]This simplifies to:\[ 3x = 5 \]

Step 2: Solve for the variable

To solve for \(x\), divide both sides by 3:\[ \frac{3x}{3} = \frac{5}{3} \]This simplifies to:\[ x = \frac{5}{3} \]

Key Concepts

Isolate the Variable

Solving linear equations often starts with the goal of getting the variable all by itself on one side of the equation. This process is known as isolating the variable. Let’s break down how we achieve this.

1. **Identify the variable**: In our initial equation, the variable we're interested in is \( x \).

• First, look at your equation: \(3x + 2 = 7\).
• The variable term here is \(3x\). Our task is to remove any numbers that are added or subtracted from this variable term.

2. **Eliminate constants**: After identifying the variable term, location of the variable, and the constant, calculate what needs to be moved.

• In our case, subtract 2 from both sides:
• Write it out as: \(3x + 2 - 2 = 7 - 2\).
• You're left with \(3x = 5\). Now the variable \(x\) is nicely isolated with only the 3 next to it.

Divide Both Sides

After isolating the variable term, the next step is to "undo" whatever multiplication or division is left on the variable. This can often simply mean dividing both sides of the equation by the same number.Let's discuss how this works in our equation.

1. **Identify the coefficient**:

• Our equation after isolation is \(3x = 5\).
• Here, 3 is the coefficient of \(x\).

2. **Divide to solve**:

• We need to remove the 3 next to \(x\). Thus, we divide both sides of the equation by 3.
• Write: \(\frac{3x}{3} = \frac{5}{3}\).
• After dividing, the equation simplifies to \(x = \frac{5}{3}\). This division step effectively frees \(x\) from its coefficient, giving us the solution.

Simplify the Equation

Simplifying an equation is a vital process in mathematics, where you rewrite it in its simplest, most efficient form. This makes equations easier to solve and understand.For our example, simplifying occurred at two main stages.

1. **After isolating**:

• Initially, we had \(3x + 2 = 7\). After isolating, we approached \(3x = 5\).
• This form is already a result of simplification, where we only deal with \(3x\) instead of the full original equation.

2. **Final simplification**:

• Once we divide both sides by 3, the equation becomes \(x = \frac{5}{3}\).
• This is the simplest form of the solution.

By simplifying, the equation is less cluttered, making it easier to see the solution clearly and confirm accuracy. This process ensures that each operation effectively simplifies the relationship between numbers and variables, leading to ease in interpretation and calculation.