Problem 9 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$y+7=21$$

Step-by-Step Solution

Answer

The solution is \( y = 14 \).

Step 1: Isolate the Variable

We begin by isolating the variable, \( y \). The equation given is \( y + 7 = 21 \). To isolate \( y \), we subtract 7 from both sides of the equation: \[y + 7 - 7 = 21 - 7\] This simplifies to \[y = 14\]

Step 2: Verify the Solution

To confirm our solution, we substitute \( y = 14 \) back into the original equation: \[14 + 7 = 21\] Simplifying the left side gives us 21, which equals the right side of the equation. Therefore, \( y = 14 \) satisfies the equation.

Step 3: Graph the Solution on a Number Line

To graph this solution on a number line, mark a point at 14, making it clear that \( y = 14 \) is the solution to the equation \( y + 7 = 21 \). You can draw a simple line and label the point '14' with a bold dot or circle.

Key Concepts

Equation Verification

After solving a linear equation, it is essential to confirm that your solution is correct. This is known as equation verification, and it involves substituting the value you've found back into the original equation. To verify the solution for the equation \( y + 7 = 21 \), we substitute \( y = 14 \). We calculate \( 14 + 7 \), which results in 21, matching the right-hand side of the equation. This indicates that our solution works and satisfies the equation perfectly. It's crucial to perform this step to ensure no calculation errors have occurred. Equation verification helps us gain confidence in our solution and ensures accuracy. Always remember: if both sides of the equation are equal after substitution, the solution is indeed verified.

Number Line Representation

Graphing solutions on a number line offers a visual way to understand and communicate the results of an equation. A number line is a horizontal line that displays numbers in a straight line format. To represent our solution \( y = 14 \), find the position corresponding to 14 on the line, and mark it with a distinct dot or circle.

This visual aid not only helps in verifying the accuracy of the solution but also provides a clear illustration of where the result stands in comparison to other numbers. It is especially helpful when dealing with inequalities or when needing to locate several solutions. Graphing on a number line is a fundamental skill that enhances comprehension and supports numerical reasoning.

Isolating Variables

When solving linear equations, isolating the variable is a crucial first step. The goal is to get the unknown variable on one side of the equation. This process often involves inverse operations. For the equation \( y + 7 = 21 \), we need to isolate \( y \). We do this by performing the inverse operation of addition, which is subtraction, on both sides of the equation.

By subtracting 7 from both sides, the equation simplifies to \( y = 14 \). This technique is versatile and applies to numerous types of equations. Understanding how to effectively isolate variables is a foundational skill in algebra, supporting the resolution of more complex mathematical problems. Always ensure that the process of isolating the variable maintains the balance of the equation by performing equal operations on both sides.