Problem 16 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$9=10+b$$

Step-by-Step Solution

Answer

The solution is \(b = -1\).

Step 1: Isolate the Variable

To solve the equation \(9 = 10 + b\), we need to isolate \(b\) by eliminating the constant term on the right side of the equation. Subtract 10 from both sides:\[9 - 10 = 10 + b - 10\]This simplifies to:\[-1 = b\]So, \(b = -1\).

Step 2: Check the Solution

To verify that \(b = -1\) is correct, substitute \(b = -1\) back into the original equation:\[9 = 10 + (-1)\]Simplify the right side:\[9 = 9\]Since both sides are equal, the solution \(b = -1\) is correct.

Step 3: Graph the Solution on a Number Line

Draw a horizontal line to represent the number line. Locate the point where the number \(-1\) is, and place a dot or mark on \(-1\) to indicate the solution. This visually represents the solution \(b = -1\).

Key Concepts

Problem-Solving

Problem-solving in mathematics involves using systematic strategies to find solutions. It's crucial to follow a logical sequence of steps. When you solve a linear equation, you're essentially finding which value of the variable makes the equation true. Here's how you usually approach solving problems like this:

• First, identify the equation you need to solve.
• Isolate the unknown variable by performing operations on both sides of the equation.
• Verify your solution by substituting it back into the original equation.
• Finally, represent the solution in a useful visual format, such as a number line.

By using these steps, you ensure your answer is correct and you've verified it visually and algebraically. Each part is critical for ensuring you fully understand the problem and the solution.

Variables

Variables are symbols in mathematics that represent numbers or values. In equations, variables allow us to express general rules that can hold for many different instances. The variable in our problem is "b". Here’s how to deal with variables:

• Recognize that a variable stands in for an unknown value that you need to find.
• Perform operations on the variable to isolate it on one side of the equation.
• Once isolated, the variable reveals the solution to the equation.

In the given exercise, the variable "b" was isolated by moving the constant 10 to the other side of the equation. Through this process, "b" was solved to be -1. Variables are essential in forming equations that describe relationships and solve problems.

Number Line

A number line is a simple visual tool that helps in understanding the magnitude and relationship of numbers. Here's how you can utilize it:

• Draw a horizontal line with evenly spaced numbers.
• Locate the solution value on the line, in this case, -1.
• Place a dot or mark on that value to represent the solution.

By using a number line, the solution becomes clear. You can easily compare and see where it lies in relation to other numbers. It's a great way to visually validate your solution and understand the placement of numbers in a tangible way. This tool is particularly useful for verifying the results of simple linear equations visually.