Problem 80 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Evaluate \(3(2 x+y)\) for \(x=1\) and \(y=5\) b. Is the number you obtained in part (a) a solution of \(4 z-30=54 ?\)

Step-by-Step Solution

Answer

a. The obtained number is 21, b. Yes, the number 21 is a solution of \(4 z-30=54\).

Step 1: Evaluate the given expression

We substitute \(x=1\) and \(y=5\) into the given expression \(3(2 x+y)\), and simplify it. \(3(2*1+5) = 3(2+5) = 3*7 = 21\)

Step 2: Test the derived result in the second equation

We substitute \(z=21\) into the equation \(4 z-30=54\) and check if it is equals to 54 or not. \(4*21 - 30 = 84 - 30 = 54\).

Step 3: Final Statement

After substituting the value \(z=21\) into the equation, it is found that it makes the equation hold true. Hence the number 21 derived from the first part is a solution to the equation \(4 z-30=54\).

Key Concepts

Substitution Method

The substitution method is one of the fundamental techniques used to evaluate algebraic expressions and solve equations in algebra. It entails replacing variables with their respective numerical values to simplify the problem at hand.

For instance, when you come across an expression like \(3(2x+y)\), and you're given specific values for \(x\) and \(y\), such as \(x=1\) and \(y=5\), your task is to replace \(x\) and \(y\) with these values. Hence, the expression becomes \(3(2*1+5)\), which simplifies to \(3(2+5)\) and then to \(3*7=21\). This process is the essence of the substitution method.

Simplifying Expressions

Simplifying expressions is a process that combines the use of arithmetic operations and algebraic properties to reduce an expression to its simplest form. This not only makes the expressions more understandable but also prepares them for further operations, like solving equations.

To simplify an expression, follow the order of operations, which is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Using our example problem, \(3(2x+y)\) simplifies by first resolving the operation inside the parentheses followed by the multiplication. This step-by-step approach ensures the expression is accurately simplified before any further algebraic steps are taken.

Solving Equations

Solving equations is an essential part of algebra involving finding the value(s) of the variable(s) that make the equation true. It often requires a variety of methods, like the substitution technique we've discussed earlier.

When solving an equation such as \(4z-30=54\), you're looking for the value of \(z\) that balances both sides of the equation. If you have already determined a possible value for \(z\) from another calculation or substitution, you can test it in this equation. In our example problem, substituting \(z=21\) yields \(4*21 - 30 = 54\), which confirms that 21 is indeed a solution as both sides of the equation are equal. This not only solves the equation but also illustrates the interplay between substitution, simplification, and the final act of solving the equation.