Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation for \(x\). $$ 4 x-3=5 $$

Step-by-Step Solution

Answer

The solution is x = 2.

Step 1: Add 3 to Both Sides

To isolate the term with the variable \(x\), we need to eliminate the constant on the left side of the equation. Add 3 to both sides of the equation: \(4x - 3 + 3 = 5 + 3\). This simplifies to \(4x = 8\).

Step 2: Divide by 4

Now, to solve for \(x\), divide both sides of the equation by 4: \(\frac{4x}{4} = \frac{8}{4}\). This gives \(x = 2\).

Key Concepts

Equation Solving

Solving an equation is like finding the key to a locked treasure chest. You have an equation, which is simply a mathematical statement that asserts the equality of two expressions. In our problem, we start with the equation \(4x - 3 = 5\). The goal is to determine the value of \(x\) that makes this equation true. To solve an equation, you perform a series of operations aimed at revealing the unknown, much like peeling layers off an onion. Each step brings us closer to uncovering \(x\).

Every equation has two sides, just like a balanced scale. Whatever you do to one side, you must also do to the other to maintain balance. Solving the equation involves keeping both sides equal while simplifying the expressions. It’s essential to follow the rules of arithmetic to ensure you're steadily approaching the solutions. Common techniques include adding or subtracting terms and multiplying or dividing them across both sides.

Algebraic Manipulation

Algebraic manipulation is the magician's trick of transforming equations into a friendlier form. In our equation \(4x - 3 = 5\), we aim to isolate terms involving \(x\).

In the first step of our solution, by adding 3 to both sides, we eliminate the pesky constant sitting beside \(4x\). This is an example of using the addition property of equality. The equation then becomes \(4x = 8\). With the constant gone, our equation looks much simpler.

This process involves strategically using arithmetic operations to rewrite the equation in a way that makes it easier to solve. The key is to decide which operations make the equation simpler step by step.

• Add or subtract to isolate terms.
• Multiply or divide to simplify coefficients.
• Follow the order of operations to keep things neat.

Algebraic manipulation is almost like puzzle-solving, each move counts toward building the final picture.

Isolating Variables

Isolating a variable is like putting a spotlight on the main actor in a play. It's about getting the variable by itself on one side of the equation. In our example, the equation \(4x = 8\) was reached by strategically eliminating other terms, so that \(x\) stands alone.

The second step in the solution involves dividing both sides by 4, the coefficient of \(x\), yielding \(x = 2\). This critical step helps uncover the specific value of \(x\) that satisfies the equation.

The principle behind isolating variables is using inverse operations. If a variable is multiplied by a number, divide by that number; if added, subtract. This untangles the variable from the equation's rest. Each move in isolating terms simplifies your equation progressively, leading you exactly to where \(x\) awaits. With practice, isolating variables transforms from a math task into an intuitive action, providing clarity and answers in diverse math scenarios.