Problem 13 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$2(4 z+3)-8=46$$

Step-by-Step Solution

Answer

The solution for \( z \) is 6, and the substitution check confirms this answer is correct.

Step 1: Distributing

Distribute the 2 to each term inside the parentheses: \(2 * 4z + 2*3 -8=46 \), which simplifies to \(8z+6-8=46\).

Step 2: Simplifying the equation

Simplify the left side of the equation: \( 8z -2= 46 \).

Step 3: Isolating Variable z

Add 2 to both sides of the equation to isolate \( 8z \) on the left side: \( 8z = 48 \). Divide both sides by 8 to solve for \( z \): \( z = 6 \).

Step 4: Substitution to Check Answer

Substitute \( z = 6 \) into the original equation to check the answer: \(2(4*6+3)-8=46\), which simplifies to \(46=46\). The left side of the equation equals the right side, and the answer is valid.

Key Concepts

Problem Solving

When solving algebraic equations, we often embark on a journey of discovery using problem-solving techniques. Our goal is to find the value of a variable that satisfies the original equation. To begin, we first assess the problem at hand, which might seem complex due to multiple operations or terms. It's essential to stay organized:

• Understand the question: What are you solving for?
• Identify given values and unknowns.
• Plan your approach to isolate and solve for the variable.

Breaking down the equation into manageable steps helps in eliminating errors and ensures a better grasp of the solution process.

Substitution Method

The substitution method is an essential tool for checking your solutions in algebra. This technique involves replacing the variable with the found solution back into the original equation to see if the equation holds true.Let's look at why substitution is important:

• Verification: It confirms the correctness of the proposed solution.
• Understanding: Reinforces the relationship between different terms in the equation.
• Confidence: Provides assurance that the solution is indeed valid.

By substituting the solution, such as substituting \(z = 6\) back into our original equation \(2(4z+3)-8=46\), and seeing if both sides result in equality, we ensure that the solution is correct.

Distributive Property

The distributive property is among the fundamental operations in algebra that simplify equations involving parentheses. It states that when you multiply a number by a sum, it is equivalent to multiplying each addend and then adding the products.To apply the distributive property correctly:

• Multiply the number outside the parentheses with each term inside. For instance, in our problem, \(2(4z+3)\) becomes \(2 * 4z + 2 * 3\).
• Simplify the resulting expression to integrate it back into the equation.
• Proceed with solving the simplified equation using standard algebraic methods.

This method assists in breaking down complex equations into simpler expressions, thereby facilitating easier and faster solving processes.