Problem 31 - College Algebra

Question

Solve for the variable. $$ 4 z-2 z(1+4)=36 $$

Step-by-Step Solution

Answer

The solution is \(z = -6\).

Step 1: Distribute the Terms Inside the Parentheses

The term \(-2z(1+4)\) can be expanded. Rewrite it as \(-2z(1+4) = -2z \times 1 + (-2z) \times 4 = -2z - 8z = -10z\).This means our equation becomes \(4z - 10z = 36\).

Step 2: Combine Like Terms

Combine the terms involving \(z\) in the equation \(4z - 10z = 36\). This simplifies to: \(-6z = 36\).

Step 3: Solve for \(z\)

Divide both sides of the equation \(-6z = 36\) by \(-6\) to isolate \(z\). This gives: \(z = \frac{36}{-6} = -6\).

Key Concepts

Algebraic Expressions

Algebraic expressions form the foundation of equations in mathematics. An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In our given problem, the expression is \(4z - 2z(1+4)\). Here, \(z\) is the variable, while 4 and -2 are coefficients. Variables are symbols, often letters, used to represent unknown values. Coefficients are numbers that multiply variables.

Expressions can be simplified by performing operations as per algebraic rules. This simplification allows for solving or transforming expressions into simpler forms, making it easier to find the value of variables. Understanding algebraic expressions is essential for tackling various problems in mathematics, especially when solving equations.

Distributive Property

The distributive property is a key algebraic property used to simplify expressions and equations. It states that the product of a number and a sum is the same as the sum of the individual products. Mathematically, this is expressed as \(a(b + c) = ab + ac\). In the given problem, we use this property to expand \(-2z(1+4)\).

Applying the distributive property, we get:

• Multiply \(-2z\) by 1, resulting in \(-2z\).
• Multiply \(-2z\) by 4, resulting in \(-8z\).

Next, add the two products to get \(-2z - 8z = -10z\).
The expanded form helps simplify the equation, making it easier to solve. The distributive property is particularly useful when dealing with parentheses in algebraic expressions.

Solving Equations

Solving equations involves finding the value of the variable that makes an equation true. Once the equation \(4z - 10z = 36\) is simplified using the distributive property and combining like terms, it becomes essential to manipulate it to isolate the variable.

Follow these steps to solve:

• First, combine like terms. Combine \(4z\) and \(-10z\) to get \(-6z\). Thus, the equation simplifies to \(-6z = 36\).
• To isolate \(z\), divide both sides by the coefficient of \(z\), which is \(-6\).
• This gives \(z = \frac{36}{-6} = -6\).

Understanding how to rearrange and solve equations is a fundamental skill in algebra. It requires careful application of mathematical operations and rules.

Negative Coefficients

Handling negative coefficients is crucial in solving algebraic equations. A coefficient is negative when it multiplies a variable by a negative number, impacting how calculations are conducted. In our problem, the expression \(-2z(1+4)\) introduces a negative coefficient through \(-2z\).

Here's how we deal with it:

• When simplifying, it's important to correctly expand and combine terms, paying attention to their signs.
• The simplification led to \(-6z = 36\), where \(-6\) is the negative coefficient of \(z\).
• To solve, divide by \(-6\). Dividing by a negative changes the sign of the result, leading to \(z = -6\).

It's essential to handle negative coefficients carefully to ensure correct solutions. Mistakes with signs can lead to incorrect answers, making attention to detail key when solving such equations.