Problem 13 - 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. \(2(4 z+3)-8=46\)

Step-by-Step Solution

Answer

The solution is \(z = 6\).

Step 1: Simplify the equation

Begin by distributing the 2 in the expression \(2(4 z+3)\) to obtain \(2 \cdot 4z + 2 \cdot3 - 8 = 46\), which simplifies to \(8z +6 -8 = 46\).

Step 2: Continue simplifying

Solve the left side of the equation by adding −2, which simplifies to \(8z = 48\).

Step 3: Solve for \(z\)

Divide both sides by 8 to isolate \(\(z\)\) and obtain our proposed solution, \(z = 6\).

Step 4: Check the solution

Substitute \(z = 6\) into the original equation \(2(4 z+3)-8=46\). This results in \(2(4 \cdot 6+3)-8=46\), which simplifies to \(46 = 46\), proving our solution to be true.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Variables, such as the \(z\) in our exercise, are symbols used to represent unknown values. The expression \(2(4z + 3)\) contains the multiplication of 2 with another expression inside the parentheses, \(4z + 3\), which is a combination of both a variable \(z\) and constants (4 and 3).

In our exercise, these constants help determine the amount \(z\) will be scaled and shifted within the equation. By understanding how to manipulate these expressions using basic arithmetic operations—addition, subtraction, multiplication, and division—you can begin to uncover the value of the variable that makes the equation true. Recognizing the parts of algebraic expressions is the first step in solving linear equations.

Equation Simplification

Simplifying an equation makes it easier to solve. The process typically involves combining like terms and removing parentheses. As shown in the solution, we use the distributive property to eliminate parentheses: \(2 \cdot 4z + 2 \cdot 3\). This property allows us to multiply 2 by each term inside the parentheses, creating a simpler expression, \(8z + 6\). Simplification is continued by subtracting 8 from both sides of the equation, leaving \(8z = 48\).

Such step-wise simplification brings us closer to isolating the variable and finding its value. It's essential to perform the same operations on both sides of the equation to keep it balanced. Equation simplification is all about making the problem more transparent and straightforward to solve.

The Substitution Method

Once a proposed solution is obtained, it must be verified, and the substitution method is the tool for this task. After simplifying and solving for the variable \(z = 6\), we substitute this value back into the original equation \(2(4z + 3) - 8 = 46\) to ensure that it satisfies the equation. Substitution involves replacing the variable with its obtained value and simplifying the expression to verify that both sides of the original equation are equal.

In our case, substituting \(z\) with 6 yields \(2(4 \cdot 6 + 3) - 8\), which simplifies to 46 on both sides, confirming that our solution is correct. Accuracy in solving is crucial; thus, this technique acts as a quality check to avoid errors and ensure that the solution makes the original equation a true statement.