Problem 22 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$y-49=-13$$

Step-by-Step Solution

Answer

The solution is \( y = 36 \).

Step 1: Isolate the Variable

To solve for \( y \), you need to isolate the variable on one side of the equation. The current equation is \( y - 49 = -13 \). To eliminate \(-49\), add \(49\) to both sides. This gives:\[ y - 49 + 49 = -13 + 49 \]Simplifying both sides will yield:\[ y = 36 \]

Step 2: Check the Solution

Substitute \( y = 36 \) back into the original equation to verify it is correct:\[ 36 - 49 = -13 \]Calculate the left side:\[ -13 = -13 \]Since both sides are equal, \( y = 36 \) is a correct solution.

Step 3: Graph the Solution on a Number Line

To graph \( y = 36 \) on a number line, draw a straight horizontal line, label a point as \(36\), and mark a dot at this point. The dot highlights the solution.

Key Concepts

Variable Isolation

When solving linear equations, the first step is often to isolate the variable. This means getting the variable, such as \( y \), by itself on one side of the equation so you can determine its value. In the problem \( y - 49 = -13 \), our goal is to solve for \( y \).

To isolate \( y \), we need to eliminate the number that's affecting it, which in this case is \(-49\). Since \(-49\) is subtracted from \( y \), we counteract it with its opposite—addition. Add \( 49 \) to both sides of the equation:

• Equation in context: \( y - 49 + 49 = -13 + 49 \)

When simplified, the left side results in \( y \), and the right side equals \( 36 \), thus giving us \( y = 36 \). By using operations, like addition or subtraction, we can systematically reduce equations until the variable stands alone. It's important to perform the same operation on both sides to maintain the balance of the equation.

Substitution Method

Once you have isolated the variable and found a solution, it's crucial to verify its correctness. This step is often overlooked, but it's essential for ensuring your solution is accurate. In our exercise, after isolating \( y \), we found \( y = 36 \).

To check this, we use the substitution method, which involves substituting the found solution back into the original equation. This confirms that the solution satisfies the equation. For our problem, substitute \( y = 36 \) into the original equation \( y - 49 = -13 \):

• Calculation: \( 36 - 49 = -13 \)

Upon simplifying, the left side equals \(-13\), matching the right side. If both sides are equal, the solution is correct. This validation step ensures you didn't make a mistake in the earlier steps and increases confidence in your solution.

Number Line Graphing

When it comes to understanding numerical solutions visually, graphing on a number line is a powerful tool. It helps you see the solution in a clear, straightforward manner. For the problem \( y = 36 \), let's graph it.

To start, draw a straight horizontal line to represent the number line. Then, mark a point labeled \( 36 \) on this line. Finally, place a dot at \( 36 \) to indicate the position of the solution:

• Visual representation: A dot precisely at the number \( 36 \).

This method allows you to visualize where the solution lies in relation to other numbers. It's a useful way to interpret the result, especially when dealing with more complex equations or systems that involve multiple solutions. Graphing enhances comprehension and presents data in an accessible form.