Problem 41 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$2 x+3 x-5=7+3$$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Simplify the Left Side of the Equation

Combine like terms on the left side. The terms with \(x\) are \(2x\) and \(3x\), which can be added together: \(2x + 3x = 5x\). The equation becomes: \(5x - 5\).

Step 2: Simplify the Right Side of the Equation

Simplify the right side of the equation by adding the constants together: \(7 + 3 = 10\). The equation now looks like: \(5x - 5 = 10\).

Step 3: Add 5 to Both Sides

To eliminate the \(-5\) on the left side, add 5 to both sides of the equation. This yields: \(5x - 5 + 5 = 10 + 5\), which simplifies to \(5x = 15\).

Step 4: Divide Both Sides by 5

To solve for \(x\), divide both sides by 5: \(\frac{5x}{5} = \frac{15}{5}\), which simplifies to \(x = 3\).

Key Concepts

Simplifying Expressions

In algebra, expressions need to be simplified to make equations easier to solve. This means transforming complex equations into simpler forms without changing their value.
A common step in simplifying expressions includes organizing all parts of the expression in a way that’s manageable. Start by identifying terms that can be combined or simplified.

• Identify like terms - terms that have the same variable component raised to the same power.
• Rearrange the expression if necessary, keeping terms with the same variables together.

This process makes it easier to perform further operations needed to solve the equation.

Combining Like Terms

One of the essential steps in simplifying expressions is combining like terms. Like terms have the same variables and the same powers, such as \(2x\) and \(3x\). When you combine these, you essentially add or subtract their coefficients, which are the numbers in front of the variables.

• Example: \(2x + 3x = 5x\)
• You combine \(2\) and \(3\) because they are coefficients of like terms \(x\).

Combining like terms creates a cleaner expression without unnecessary repetition, and it simplifies further calculations.

Basic Algebra Steps

Solving a linear equation involves several fundamental algebra steps. These steps ensure that you systematically isolate the variable to determine its value.

• Step 1: Simplify each side of the equation separately.
• Step 2: Use inverse operations to move all terms containing the variable to one side and all constant terms to the other side.
• Step 3: Adjust the equation step-by-step, cross-checking as you go.

These basic steps are crucial as they guide you through solving the equation systematically and accurately.

Isolating the Variable

Isolating the variable is the final goal when solving an equation. This involves getting the variable by itself on one side of the equation, typically the left side.
To do this, you'll often need to perform operations that 'undo' others, such as:

• Addition/Subtraction: If a term is subtracted or added, do the opposite operation to isolate the variable (e.g., eliminate a \(-5\) by adding \(5\).
• Multiplication/Division: If a term is multiplied or divided, use the inverse operation to remove the coefficient (e.g., divide by \(5\) if the variable is multiplied by \(5\).

These operations gradually peel away layers surrounding the variable, uncovering its value.