Problem 31 - College Algebra

Question

For the following exercises, solve for the variable. $$ 4 z-2 z(1+4)=36 $$

Step-by-Step Solution

Answer

\( z = -6 \)

Step 1: Distribute the Terms

Start by distributing the term inside the second part of the equation: \[ 4z - 2z(1 + 4) = 36 \]which simplifies to:\[ 4z - 2z - 8z = 36 \]

Step 2: Combine Like Terms

Combine the like terms on the left side of the equation:\[ 4z - 2z - 8z = 36 \]simplifies to:\[ -6z = 36 \]

Step 3: Solve for \( z \)

To isolate \( z \), divide both sides of the equation by -6:\[ z = \frac{36}{-6} \]Therefore,\[ z = -6 \]

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable that makes the equation true, meaning both sides of the equation are equal. In a linear equation, every term is either a constant or a product of a constant and a single variable. A basic linear equation looks like this:

• \( ax + b = c \)

where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable.
To solve the equation, we manipulate it using algebraic operations to isolate \( x \) on one side of the equation. We aim to express \( x \) in terms of the other constants, if there are any, or find it as a solitary constant.

Consistency in balancing both sides is key. Whatever you do to one side of the equation, you must do to the other. Doing so maintains the equality, which is crucial for correctly solving these equations.

Variable Isolation

Variable isolation is a fundamental step in solving equations. It involves rearranging the equation so that the variable is by itself on one side.
This process often requires a few algebraic actions:

• Addition or Subtraction: Undo constant terms by using their opposite operations to shift them to the other side of the equation.
• Multiplication or Division: Isolate the variable by eliminating coefficients, ensuring the variable remains on one side alone.

Let's illustrate this with a simple example. Consider equation \( 2x + 5 = 15 \). Subtracting 5 from both sides yields \( 2x = 10 \).
Then, divide both sides by 2 to find \( x = 5 \).
The variable \( x \) is now isolated, giving us the solution directly.

Distributive Property

The distributive property is a valuable algebraic principle used in simplifying expressions and solving equations. It states that multiplying a sum by a number equals multiplying each addend by that number and then adding the products. Mathematically, it looks like this:

• \( a(b + c) = ab + ac \)

In our exercise, the distributive property was applied to simplify the term \(-2z(1 + 4)\).
If we apply the distributive property here, it becomes \( -2z \times 1 + -2z \times 4 \), which simplifies the expression to \( -2z - 8z \).
This simplification allows us to combine like terms more easily and proceed with solving the equation.
Using the distributive property can often make large expressions more manageable, leading to simpler steps in the solution process.