Problem 15 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$8(x+5)-6=18$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -2\).

Step 1: Distribute the 8

The first step is to distribute the 8 inside the parentheses. This means we need to multiply 8 by each term inside the parentheses (i.e., 8 times both \(x\) and 5). The equation becomes: \[8x + 40 - 6 = 18\].

Step 2: Simplify the Equation

Simplify the left-hand side of the equation by combining the constants. Subtract 6 from 40: \[8x + 34 = 18\].

Step 3: Isolate the Variable Term

Subtract 34 from both sides of the equation to isolate the term with variable \(8x\). This is done to cancel out the constant addition on the left. \[8x = 18 - 34\]. Simplifying the right-hand side gives us \[8x = -16\].

Step 4: Solve for x

Divide both sides of the equation by 8 to find the value of \(x\). \(x = \frac{-16}{8}\). Simplifying the fraction gives \(x = -2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify equations. It's like sharing something fairly among all terms inside a set of parentheses. When you see an expression like \(8(x + 5)\), you are asked to distribute or "spread out" the multiplication of 8 to both terms inside the parentheses, which are \(x\) and 5.

• To distribute, multiply the number outside the parenthesis by each term inside.
• In our equation, you calculate \(8 \times x\) and \(8 \times 5\).

By distributing, the equation \(8(x + 5)\) transforms into \(8x + 40\). This step helps in removing the parenthesis and breaking the equation into smaller, easier to handle parts.
Understanding and using the distributive property helps to simplify complex expressions and is a stepping stone in solving equations efficiently.

Solving Equations

Solving equations is about finding the unknown value that makes the statement true. It can be seen as a mystery where you're trying to balance both sides of the equation until you unveil the variable's value. This often requires several steps, including distributing numbers, combining like terms, and isolating the variable.

• Start with expanding and simplifying each side of the equation with the distributive property.
• Combine all like terms, which are numbers or variables you can add or subtract.

After applying the distributive property, continue by simplifying the overall expression. In the given example, simplifying \(8x + 40 - 6\) to \(8x + 34\) is crucial. You are reducing the equation to fewer terms, gradually making it simpler to understand and solve. This process helps in identifying what the variable equals so you can find your solution.

Variable Isolation

Variable isolation is the process of removing all other numbers and coefficients from the variable to see what it truly equals. This is your goal in equalities: unveil the mystery of which number the variable stands for.
To isolate a variable:

• Move all constants to the other side of the equation by performing inverse operations (e.g., subtraction if you see addition).
• In our equation \(8x + 34 = 18\), subtract 34 from both sides to get \(8x = -16\).
• Divide the resulting coefficient (in front of the variable) to uncover the variable. Divide both sides of \(8x = -16\) by 8 to solve for \(x\).

This isolation allows for the clear conclusion of \(x = -2\). It's a vital end-step in solving linear equations, ensuring the variable stands alone, giving you the precise solution. Each arithmetic operation heightens your grip on the problem, sharpening your skills in algebra. By mastering these techniques, finding unknowns becomes a straightforward and satisfying process.