Problem 48 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$4 x+8 x-2 x=15-10$$

Step-by-Step Solution

Answer

x = \(\frac{1}{2}\)

Step 1: Simplify the Left Side

The left side of the equation is \(4x + 8x - 2x\). Combine like terms by adding and subtracting the coefficients of \(x\):\[4x + 8x - 2x = (4 + 8 - 2)x = 10x\]So, the left side simplifies to \(10x\).

Step 2: Simplify the Right Side

The right side of the equation is \(15 - 10\). Simplify by performing the subtraction:\[15 - 10 = 5\]Thus, the right side simplifies to \(5\).

Step 3: Set the Simplified Expression

Now, set the simplified left side equal to the simplified right side:\[10x = 5\]

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation by 10:\[x = \frac{5}{10} = \frac{1}{2}\]Thus, \(x\) is \(\frac{1}{2}\).

Key Concepts

Like Terms

In mathematics, when dealing with algebraic expressions, you often encounter the term "like terms." Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(4x + 8x - 2x\), all terms are like because they each contain the variable \(x\) raised to the first power.

Combining like terms is a fundamental step in simplifying algebraic expressions. It involves adding or subtracting the coefficients (the numerical part of the term) while keeping the variable part unchanged.

• Example: In \(4x + 8x - 2x\), you add together \(4\), \(8\), and \(-2\) to get \(10\), resulting in \(10x\).
• This process makes complex expressions more manageable, allowing you to work with simpler forms.

Practically, combining like terms simplifies equations and is the stepping stone for the further solving process. Always ensure terms share the exact variable and exponent to be considered like terms.

Simplification

Simplification is the process of transforming an equation or expression into its simplest form. This is often the first step when solving equations, as it clears up any unnecessary complexity.

The goal of simplification is to make equations easier to handle and solve, by breaking down more complex expressions into simpler components.

• On the left side of the equation \(4x + 8x - 2x\), simplification involves combining like terms. This boils it down to \(10x\).
• On the right side, simplification involves arithmetic, like \(15 - 10\), simplifying it to \(5\).

Simplifying both sides prevents errors that might occur from dealing with larger expressions. It also highlights the structure of an equation, aiding in the identification of the main operation needed to solve it.

Equation Solving Steps

Once the sides of an equation are simplified, the next task is to efficiently solve the equation. Solving equations typically follows a series of logical steps that bring you to the answer.

For the given exercise, after simplification, you have \(10x = 5\). To solve for \(x\), you need to isolate the variable on one side.

• Step 1: Identify the simplified equation, which is \(10x = 5\).
• Step 2: Divide both sides by the coefficient of \(x\), which is \(10\), to isolate \(x\). This gives \(x = \frac{5}{10}\).
• Step 3: Simplify the fraction \(\frac{5}{10}\) to \(\frac{1}{2}\). Thus, \(x\) is \(\frac{1}{2}\).

Following these steps ensures you arrive at the correct solution systematically. Each action, whether subtracting, dividing, or simplifying, moves you closer to solving for the variable.