Problem 4 - Prealgebra

Question

Solve each equation. Check your solution. $$16(z+3)=4(z+9)$$

Step-by-Step Solution

Answer

The solution is \( z = -1 \).

Step 1: Distribute

Distribute the numbers outside the parentheses to the terms inside them. For the left side, multiply 16 by each term inside the parenthesis: \[ 16(z+3) = 16z + 48 \] For the right side, multiply 4 by each term inside the parenthesis:\[ 4(z+9) = 4z + 36 \]

Step 2: Set Up Equation

After distribution, the equation becomes:\[ 16z + 48 = 4z + 36 \]

Step 3: Isolate the Variable

Subtract \( 4z \) from each side to move all \( z \) terms to one side:\[ 16z - 4z + 48 = 36 \] which simplifies to:\[ 12z + 48 = 36 \] Next, subtract 48 from each side to isolate the \( z \) term:\[ 12z = 36 - 48 \] which further simplifies to:\[ 12z = -12 \]

Step 4: Solve for the Variable

Divide each side by 12 to solve for \( z \):\[ z = \frac{-12}{12} \] \[ z = -1 \]

Step 5: Check Your Solution

Substitute \( z = -1 \) back into the original equation to verify the solution:\[ 16(-1 + 3) = 4(-1 + 9) \]Simplify both sides:- Left side: \[ 16(2) = 32 \] - Right side: \[ 4(8) = 32 \]Since both sides are equal when \( z = -1 \), the solution is verified.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical statement that includes numbers, variables, and operation signs. In the equation \(16(z+3)=4(z+9)\), there are two algebraic expressions:

• The left-side expression is \(16(z+3)\)
• The right-side expression is \(4(z+9)\)

Each expression consists of a coefficient (16 or 4) multiplyied by the term inside the parenthesis, which includes both a variable \(z\) and a constant. To handle an algebraic expression effectively, you need to understand how to manage the variables and constants together by using operations such as distribution or combining like terms. This forms the groundwork for solving an equation by balancing both sides.

Mastering Variable Isolation

The purpose of isolating the variable is to get the variable alone on one side of the equation. Our ultimate goal is to solve for \(z\) and know its value. In the exercise, you start by working through several steps to shift all terms containing the variable to one side:

• Move terms using subtraction or addition, like subtracting \(4z\) from both sides to consolidate all \(z\) terms: \(16z - 4z + 48 = 36\)
• Further simplify: \(12z + 48 = 36\)
• Next, subtract 48 from both sides to isolate the \(z\) terms: \(12z = 36 - 48\), which simplifies to \(12z = -12\)
• Divide by the coefficient of \(z\) to solve: \(z = \frac{-12}{12}\), giving \(z = -1\)

By doing this, you will clearly see the value of \(z\), ensuring you've successfully isolated the variable.

Importance of Equation Verification

After finding the solution of an equation, verifying your solution ensures that no mistakes were made. Verification involves substituting the solution back into the original equation to check if both sides remain balanced. For example, take \(z = -1\) back to the original equation \(16(z+3) = 4(z+9)\):

• Substitute \(z = -1\) into both expressions: \(16(-1 + 3)\) and \(4(-1 + 9)\)
• Simplify each side: Left becomes \(16(2) = 32\) and Right becomes \(4(8) = 32\)
• Since both sides match as \(32 = 32\), the solution \(z = -1\) is verified

Verification is a crucial step which confirms the accuracy of your solution, reinforcing both your understanding and confidence in solving equations.