Problem 15 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$y+8=3$$

Step-by-Step Solution

Answer

The solution is \( y = -5 \).

Step 1: Isolate the Variable

Start by isolating the variable \( y \) in the equation \( y + 8 = 3 \). To do this, subtract 8 from both sides of the equation. This gives: \[ y + 8 - 8 = 3 - 8 \] which simplifies to \[ y = -5 \].

Step 2: Verify the Solution

To ensure the solution is correct, substitute \( y = -5 \) back into the original equation \( y + 8 = 3 \). Substitute to check: \[ -5 + 8 = 3 \], which simplifies to \( 3 = 3 \). The original equation holds true, confirming that the solution is correct.

Step 3: Graph on Number Line

To graph the solution, locate the point \( y = -5 \) on a number line. Mark \( -5 \) with a distinct point or dot to indicate that it is the solution to the equation.

Key Concepts

Isolating Variables

Isolating variables is a key step in solving equations. It involves getting the unknown variable, often represented as a letter, by itself on one side of the equation. This process makes it easier to see what the variable equals. To isolate the variable, you need to perform the same operations on both sides of the equation to ensure it remains balanced. In the exercise, for instance, the equation is given as \( y + 8 = 3 \). To isolate \( y \), you subtract 8 from both sides, which results in \( y = -5 \). This step is crucial, as it directly leads you to the solution of the equation. Remember, the aim is to perform operations that cancel out everything except the variable on one side, while maintaining equality.

Verifying Solutions

Verifying solutions is about making sure your answer is correct. Once you've solved the equation and found a potential solution, substitute it back into the original equation to see if it works. In our case, after determining that \( y = -5 \), we check if substituting \( -5 \) into the original equation \( y + 8 = 3 \) holds true.

• Substitute \( -5 \) for \( y \): \( -5 + 8 = 3 \).
• Perform the addition: \( 3 = 3 \).

This confirms the left side equals the right side of the equation, verifying the solution is correct. Always performing this check can help catch any mistakes and assure accuracy in your calculations.

Graphing on a Number Line

Graphing solutions on a number line provides a visual representation of where the solution lies. For instance, with the equation \( y = -5 \), you would locate \( -5 \) on a number line.

• Draw a number line and mark equal increments.
• Identify the point \( -5 \) and highlight it with a distinct dot or mark.

This makes the solution not just a number, but also a visual point on a line, helping to better understand its position relative to other numbers. Graphing can especially aid in comparing solutions or handling more complex inequalities or systems of equations.