Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. Eight subtracted from the product of 4 and a number is 56

Step-by-Step Solution

Answer

The solution for \(x\) in the equation derived from the problem is 16.

Step 1: Formulate the equation

The statement 'eight subtracted from the product of 4 and a number' can be translated into an equation as \(4x - 8 = 56\). The operation 'the product of 4 and a number' is represented by \(4x\) and 'eight subtracted from' is represented by subtraction of 8.

Step 2: Solve for \(x\)

You first add 8 to both sides of the equation \(4x - 8 = 56\) to get \(4x = 64\). Then divide both sides by 4 to isolate \(x\), yielding \(x = 16\).

Step 3: Verify the solution

Substitute \(x = 16\) in the original equation \(4x - 8 = 56\). It simplifies as \(4*16 - 8 = 56\), which indeed equals 56, satisfying the equation. Therefore, the solution is correct.

Key Concepts

Solving equations

When you are solving equations, you are finding the value of the variable that makes the equation true. In our example equation, \(4x - 8 = 56\), we're trying to find the number represented by \(x\). To solve equations like this one, follow these steps:

• Identify the operations that are being performed on the variable (in this case, multiplication by 4 and subtraction of 8).
• Choose the inverse operations to undo these effects, moving step-by-step to isolate the variable on one side of the equation.

This process helps in systematically breaking down the equation into simpler terms until you find \(x\). Remember, whatever you do to one side of the equation, you must do to the other to maintain balance.

Linear equations

Linear equations are a type of equation where the highest power of the variable is 1. Essentially, they form a straight line when you graph them. In algebra, they appear in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. Our equation, \(4x - 8 = 56\), is a typical linear equation:

• The term with the variable \(4x\) signifies linear growth or scaling by 4.
• The \(-8\) and \(56\) are constants adjusting that growth.

Linear equations are one of the most straightforward types of equations to solve because they involve simple arithmetic operations and follow predictable patterns. They are the foundation for understanding more complex algebraic concepts.

Problem-solving steps

Approaching a math problem systematically helps you find the correct solution efficiently. For equations, follow a series of problem-solving steps:

• Understand the problem: Break down the statement into smaller parts, identifying what is given and what is to be found.
• Translate into mathematical terms: Convert the verbal statement into an algebraic equation.
• Apply operations step-by-step: Use inverse operations to simplify and solve the equation.
• Check your work: After solving, always re-evaluate the problem to ensure accuracy.

By following these steps, you can approach algebra problems with a clear strategy, reducing errors and increasing confidence in solving equations.

Verify solutions

Once you solve an equation, it's crucial to verify your solution to confirm it's correct. This verification step acts as a proof that your method was sound and accurate. In our example:

• Take your solution \(x = 16\) and substitute it back into the original equation \(4x - 8 = 56\).
• Calculate: \(4 \cdot 16 - 8\) equals 64 - 8, which simplifies to 56.
• Since the left side equals the right side, \(56 = 56\), the solution is verified.

By taking this extra step, you double-check your work, ensuring the answer solves the equation correctly and adheres to the problem's constraints.