Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(3(3 z+5)-7=89\)

Step-by-Step Solution

Answer

The solution of the equation is \(z = 9\). After substituting this back into the original equation, it holds true and thus is the correct solution.

Step 1: Simplify the equation

Initially, simplify the equation by expanding the expression on the left side of the equation. Distribute the 3 into the parentheses: \(3 * 3z + 3 * 5 - 7 = 89\) which simplifies to \(9z + 15 - 7 = 89\).

Step 2: Solve for 'z'

Combine like terms on the left side to get \(9z + 8 = 89\). Now, isolate 'z' by first subtracting 8 from both sides: \(9z = 89 - 8\). Then, divide both sides by 9 to solve for 'z': \(z = 81/9\).

Step 3: Check the solution

Substitute the proposed solution back into the original equation: \(3(3 * 9 + 5) - 7 = 89\). Evaluate the expression and confirmation of a correct solution is when the left side equals the right side of the equation.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that plays a crucial role in solving equations involving parentheses. When you see an expression like \(3(3z + 5)\), the distributive property allows you to expand this expression by multiplying the term outside the parentheses with each term inside. For instance, in this problem, you will multiply 3 by both \(3z\) and 5, resulting in \(3 * 3z + 3 * 5\). This gives us \(9z + 15\). Using the distributive property simplifies expressions and helps us to eliminate parentheses, turning them into a format that makes other algebraic manipulations, like combining like terms, much easier. It's a key step in transitioning an equation into a simpler form which can be solved for the variable.

Isolating the Variable

Isolating the variable means getting the variable you are solving for, all by itself on one side of the equation. In this equation, we start with \(9z + 8 = 89\). The goal is to solve for \(z\). First, we need to perform actions that help isolate \(z\). Start by subtracting 8 from both sides of the equation. This step ensures the equation remains balanced and helps move us closer to having \(z\) isolated:

• Initial equation: \(9z + 8 = 89\)
• After subtracting 8: \(9z = 81\)

Next, divide both sides by 9, to get \(z\) by itself:

• Resulting equation: \(z = 81 / 9\)
• Simplified to: \(z = 9\)

This step completes the isolation of \(z\), giving us the solution to the equation. Isolating the variable is crucial because it gives us the answer we are looking for without ambiguity.

Checking the Solution

After solving an equation, the final important step is to check your solution. This ensures that the solution you found indeed satisfies the original equation. For this problem, we substitute \(z = 9\) back into the original equation: \(3(3z + 5) - 7 = 89\). Replace \(z\) with 9:

• Step one: \(3(3 * 9 + 5) - 7\)
• Simplify inside the parentheses: \(3(27 + 5) - 7\)
• Continue simplifying: \(3(32) - 7\)
• Multiply: \(96 - 7\)
• Final result: 89

Since the left side equals the right side of the original equation, the solution \(z = 9\) is verified as correct. Checking solutions is an essential step to catch any possible errors made during the solving process, ensuring your answer is not just a solution, but the correct solution.