Problem 48 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$5 z=30 ; 8$$

Step-by-Step Solution

Answer

No, 8 is not a solution to the equation \(5z = 30\).

Step 1: Understand the Equation

Understand that the equation \(5z = 30\) implies that a number \(z\) multiplied by \(5\) equals \(30\).

Step 2: Substitute the Value

Substitute the given number, \(8\), for \(z\) in the equation. This means replace \(z\) with \(8\) to get \(5*8 = 30\).

Step 3: Solve the Equation

Solve the left side of the equation. Here, \(5*8\) gives \(40\), not \(30\). This means that \(8\) is not a solution to the equation \(5z = 30\).

Key Concepts

Solving Equations

Solving an equation involves finding the value or values that make the equation true. An equation like \(5z = 30\) means that we are looking for a number \(z\) that, when multiplied by 5, equals 30. This process often involves manipulating the equation to isolate the variable on one side. The goal is to perform operations that simplify the equation while keeping both sides equal. For simple linear equations like \(5z = 30\), you can divide both sides by 5 to solve for \(z\). When you do this, you find that \(z = \frac{30}{5} = 6\). This tells you that 6 is the number that satisfies the equation, meaning it is the correct solution.

Algebraic Substitution

Algebraic substitution is a key method in solving equations and involves replacing a variable with a given number. In our exercise, we substitute 8 for \(z\) in the equation \(5z = 30\). This gives us \(5 \times 8 = 30\). It's like checking if our substitute value makes the original equation hold true. After performing the multiplication, we get 40 on the left side. This result indicates that 8 does not satisfy the equation because 40 is not equal to 30. Through substitution, we test if a number is a solution by seeing if it maintains the equality of the equation.

Solution Verification

Solution verification ensures that your calculated answer is correct. In algebra, this involves checking your work by substituting the solution back into the original equation to see if it holds true. Suppose you suspect a number is a solution to an equation; you substitute it back into the equation. If both sides are equal, you've verified the solution is correct. In our example, after substitution, we found \(5 \times 8\) resulted in 40. Since 40 does not equal 30, we verified that 8 is not a correct solution. Verifying is an essential step in solving equations to ensure accuracy and validate your answer.