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Chapter 3: Problem 1

Solve each equation. Check your solution and graph it on a number line. $$x+14=25$$

Short Answer

Expert verified
The solution is \( x = 11 \).

Step by step solution

01

Identify the Equation

We need to solve the equation given: \( x + 14 = 25 \). Our goal is to find the value of \( x \) that satisfies this equation.
02

Isolate the Variable

To find \( x \), we need to isolate it by performing the inverse operation. Subtract 14 from both sides of the equation:\[x + 14 - 14 = 25 - 14\]This simplifies to:\[x = 11\]Now we have found \( x = 11 \).
03

Check the Solution

Substitute \( x = 11 \) back into the original equation to verify:\[11 + 14 = 25\]Simplifying the left side gives:\[25 = 25\]The equation is true, confirming \( x = 11 \) as the correct solution.
04

Graph the Solution on a Number Line

Draw a number line and locate the point corresponding to \( x = 11 \). Mark this point clearly, as it represents the solution to the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Checking Solutions
After solving an equation, it's crucial to check that the solution is correct. This ensures precision and accuracy, because mistakes can happen even in simple calculations. To check a solution:
  • Substitute the solution back into the original equation.
  • Perform the arithmetic operations to see if both sides of the equation equal each other.
For example, once we found that the solution to the equation \( x + 14 = 25 \) was \( x = 11 \), we plugged 11 back into the equation.
We calculated \( 11 + 14 \), which indeed equals 25. This verifies that our solution is correct and shows that verifying solutions is a key step in problem-solving.
Graphing on a Number Line
Graphing solutions on a number line helps visualize mathematical answers. It gives a clear picture of the value's position in relation to other numbers. Here's how to plot a solution:
  • Draw a horizontal line and mark regular intervals for integers.
  • Locate the point that corresponds to the solution.
  • Use a dot or circle on the number line to mark this point.
In our exercise, we found \( x = 11 \). On a number line, we place a dot above 11.
Number lines are helpful, especially for visual learners, as they represent the solution plainly and ensure understanding of where the number lies in the numerical system.
Inverse Operations
Inverse operations allow us to isolate variables and solve equations effectively. They are operations that undo each other and are crucial in algebraic problem-solving. Common inverse operations include:
  • Addition and subtraction
  • Multiplication and division
To solve the equation \( x + 14 = 25 \), we use the inverse operation of addition, which is subtraction, to isolate \( x \). By subtracting 14 from both sides, we simplify the equation to \( x = 11 \).
Inverse operations make it possible to "reverse" actions in equations, revealing the value of the variable. Understanding and applying these operations correctly is essential for solving all kinds of equations efficiently.

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