Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$5 x-4=2(x-2)$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Expand the equation

First, we need to expand the right-hand side of the equation. Start by distributing the 2 across the terms inside the parentheses:\[ 2(x-2) = 2x - 4 \]Replace the right-hand side in the original equation with this expansion:\[ 5x - 4 = 2x - 4 \]

Step 2: Move terms with x to one side

Now, let's move all terms involving \( x \) to one side of the equation. Subtract \( 2x \) from both sides of the equation:\[ 5x - 2x - 4 = -4 \]This simplifies to:\[ 3x - 4 = -4 \]

Step 3: Isolate the term with x

Add 4 to both sides of the equation to move the constant term away from the left:\[ 3x - 4 + 4 = -4 + 4 \]This simplifies to:\[ 3x = 0 \]

Step 4: Solve for x

Finally, divide both sides by 3 to solve for \( x \):\[ \frac{3x}{3} = \frac{0}{3} \]This simplifies to:\[ x = 0 \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that allows us to find the value of a variable that makes an equation true. The basic idea is to manipulate the equation until the variable is isolated on one side. This process typically involves several steps, such as simplifying terms, collecting like terms, and adjusting both sides of the equation equally to maintain balance.

• Start from the original equation and follow procedural steps.
• Maintain equilibrium by performing the same operations on both sides.
• Work using basic arithmetic operations such as addition, subtraction, multiplication, and division.

By understanding and practicing these steps, you become proficient at solving various types of algebraic equations, allowing you to tackle more complex problems confidently.
While mathematics might seem complex, breaking down problems into systematic steps makes them easier to solve.

Algebraic Expressions

Algebraic expressions are like building blocks in algebra. They consist of numbers, variables, and arithmetic operators. Understanding what an algebraic expression represents is crucial for manipulating equations. For example, in the given equation, we observe algebraic expressions on both sides:

• The left-hand side: \(5x - 4\)
• The right-hand side: \(2(x-2)\)

It's important to recognize the role of each part of these expressions.
The coefficient is the number multiplied by the variable, like the 5 in \(5x\). The constant is the number without a variable, like 4 in the expression \(-4\). Expanding expressions by using distribution, as done in the solution, simplifies and reveals their full form, making further manipulation possible.
Working with algebraic expressions is a skill that enables you to rewrite and interpret equations, setting you up to solve for variables.

Equation Manipulation

Equation manipulation involves rearranging and simplifying an equation to isolate the variable in question. Through a series of strategic moves, we reshape the equation until it reveals the solution. This process includes:

• Distributing terms to remove parentheses and simplify the equation, as seen in the original solution.
• Combining like terms to streamline the equation and reduce complexity.
• Shifting terms across the equal sign, making sure to change their signs appropriately to keep the equation balanced.

In the original solution, for instance, equation manipulation involved moving all terms involving the variable to one side (like when we subtracted \(2x\) from both sides) and then isolating the variable by removing constants (as when we added 4 to both sides). Finally, we solved for \(x\) by dividing by the coefficient of \(x\).
Mastering the art of equation manipulation is like solving a puzzle, where each move is carefully calculated to progressively get you closer to the "solved" form of the equation.