Problem 3 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$16=y+20$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = -4\) and it checks out when verified.

Step 1: Identify the Equation

The equation given is \(16 = y + 20\). This is a simple linear equation where we need to solve for \(y\).

Step 2: Isolate the Variable

To solve for \(y\), we need to isolate it on one side of the equation. We can do this by subtracting 20 from both sides of the equation. This gives us:\[ 16 - 20 = y + 20 - 20 \]Simplifying both sides, we get:\[ -4 = y \]

Step 3: Verify the Solution

To verify the solution, substitute \(-4\) back into the original equation and check if both sides are equal:\[ 16 = (-4) + 20 \]\[ 16 = 16 \]Since both sides are equal, \(y = -4\) is the correct solution.

Step 4: Graph Solution on Number Line

To graph the solution on a number line, mark a point at \(-4\). This represents the solution to the equation, indicating where \(y\) is located on the number line.

Key Concepts

Solving Equations

Solving linear equations is a fundamental skill in algebra that allows us to find the value of unknown variables. In the given equation, \(16 = y + 20\), our goal is to determine what value of \(y\) would make the equation true. To achieve this, we need to perform operations that simplify the equation, essentially "undoing" any addition or subtraction present. By rearranging the equation, we ensure the variable is alone on one side, which helps us identify its value easily. This process also involves checking our work by substituting the found variable back into the original equation to ensure both sides remain equal. This ensures accuracy and reinforces the correct solution.

Variable Isolation

In linear algebra, the process of variable isolation is crucial to solving equations. Variable isolation means transforming an equation so that the variable we are interested in is all by itself on one side of the equation. With our equation \(16 = y + 20\), we aim to isolate \(y\). We do this by subtracting 20 from both sides:

• Start with: \(16 = y + 20\)
• Subtract 20 from each side: \(16 - 20 = y + 20 - 20\)
• Simplify: \(-4 = y\)

Now the variable \(y\) is isolated, showing us \(y = -4\). This step-by-step subtraction method is essential to unravel and solve many algebraic problems, as it provides a clear path to finding the unknown.

Number Line Graphing

Graphing on a number line provides a visual representation of our solution, illustrating where a number fits in relation to others. After solving the equation \(-4 = y\), we know that \(y = -4\). To show this on a number line:

• Draw a straight line and mark evenly spaced intervals for integers around
• Focus on the region where \(-4\) is located
• Place a point or dot directly above or on the number \(-4\) to represent the solution

This graph highlights the position of \(y\) within the context of its surrounding numbers and can aid in understanding numerical relationships and the solution's significance.