Problem 25 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$23+y=14$$

Step-by-Step Solution

Answer

The solution is \( y = -9 \).

Step 1: Isolate the Variable

The equation we have is \( 23 + y = 14 \). Our goal is to solve for \( y \). To do this, subtract 23 from both sides of the equation:\[23 + y - 23 = 14 - 23\]Simplifying both sides, we get:\[y = -9\]

Step 2: Check the Solution

To verify our solution, substitute \( y = -9 \) back into the original equation and check the result:\[23 + (-9) = 14\]Simplify the left side:\[23 - 9 = 14\]After simplifying, we indeed get:\[14 = 14\]Since both sides of the equation are equal, \( y = -9 \) is the correct solution.

Step 3: Graph the Solution on a Number Line

Draw a horizontal line representing a number line. Place markings for numbers at equal intervals and label them. Identify the position corresponding to \( y = -9 \) on this number line, and draw a point or a bolded line over it to represent the solution.

Key Concepts

Isolating Variables

In mathematics, isolating the variable is a key step in solving equations. This means getting the variable you are solving for on one side of the equation by itself. In this exercise, the given equation is \(23 + y = 14\). The goal is to solve for \(y\). To isolate \(y\), you need to peel away any numbers or terms attached to it.

Start by performing operations that remove these numbers. Here, we subtract 23 from both sides of the equation:

• On the left side: \(23 + y - 23\)
• On the right side: \(14 - 23\)

Simplifying both sides, you end up with \(y = -9\). This step is crucial because it simplifies the equation to a form where \(y\) is isolated, making it easier to understand what value \(y\) represents in satisfying the equation.

Checking Solutions

After finding the value of the variable, it is important to check the solution. This ensures that the value truly satisfies the original equation. Substituting \(y = -9\) back into the original equation \(23 + y = 14\) allows you to verify the accuracy:

• Replace \(y\) with \(-9\): \(23 + (-9) = 14\)
• Simplify the expression: \(23 - 9 = 14\)

Once you simplify both sides, you observe that \(14 = 14\). This confirms that the solution \(y = -9\) is indeed correct, as the left side equals the right side. Checking helps especially when mistakes might have been made in calculation or algebraic manipulation.

Graphing Solutions on Number Line

Graphing a solution on a number line provides a visual representation of the solution. To graph \(y = -9\), start by drawing a horizontal line for the number line.

Mark numbers at equal intervals along the line, including \(-9\). Once the line is labeled, locate \(-9\) and draw a point directly above it to mark the solution. This bolded point or mark signifies where \(y = -9\) fits on the number line. Visual representations like this aid in understanding and verifying that \(-9\) is the correct value for \(y\) in relation to other numbers.

Integer Operations

Understanding and performing integer operations is foundational in managing the steps needed to solve equations. In this exercise, subtraction was used to isolate the variable, demonstrating integer operations in action. Let's break it down:

• Subtraction Operation: To isolate \(y\), subtract 23 from both sides. Integer subtraction is a basic operation where you subtract one number from another. Here, \(23\) is subtracted from \(14\) which results in a negative value: \(-9\).
• Handling Negative Numbers: By understanding the rules of integer operations, you know that subtraction leading to a negative number is expected when removing a larger integer than what you start with.

Mastering these integer manipulations allows for the smooth isolation of variables and simplification of equations, critical skills in algebra.