Problem 23 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$-15=x-16$$

Step-by-Step Solution

Answer

The solution is \(x = 1\). This solution is verified and can be graphed as a point at 1 on the number line.

Step 1: Isolate the Variable

The first step is to solve the equation \(-15 = x - 16\) to isolate \(x\). To do this, we need to get \(x\) by itself on one side of the equation. To eliminate the \(-16\) on the right side, we add \(16\) to both sides of the equation: \(-15 + 16 = x - 16 + 16\). This simplifies to \(x = 1\).

Step 2: Verify the Solution

To ensure our solution is correct, we substitute \(x = 1\) back into the original equation. Replace \(x\) in the original equation with \(1\): \(-15 = 1 - 16\). Simplifying \(1 - 16\), we get \(-15\), which matches the left side of the equation. Therefore, \(x = 1\) is correct.

Step 3: Graph the Solution on a Number Line

To represent the solution on a number line, draw a horizontal line representing all possible values for \(x\). Since our solution is \(x = 1\), indicate this point on the number line with either a point or a circle around the number 1 on the number line.

Key Concepts

Isolating Variables

When solving linear equations, the primary goal is to isolate the variable, often represented as \(x\). This means getting \(x\) by itself on one side of the equation.
Consider the equation \(-15 = x - 16\). Here, \(x\) is subtracted by 16, so to isolate \(x\), we need to get rid of the \(-16\) on the right side. We do this by performing the inverse operation, which is adding 16 to both sides of the equation.
Here's how it works:

• Add 16 to both sides:
  \(-15 + 16 = x - 16 + 16\)
• This simplifies to:
  \(x = 1\)

By ensuring the variable is isolated, we can plainly see the value that solves the equation.

Verifying Solutions

Once you have isolated the variable and found a potential solution, it is crucial to verify this solution for accuracy. Verification involves plugging the found value back into the original equation and checking if both sides are equal.
Let's verify \(x = 1\) as follows:

• Substitute \(x = 1\) back into the original equation:
  \(-15 = 1 - 16\)
• Simplify the right side:
  \(1 - 16 = -15\)
• Observe that both sides of the equation are indeed equal:
  \(-15 = -15\)

This confirms that our solution, \(x = 1\), is correct. Verification is a critical step that ensures no mistakes were made during isolation and arithmetic simplification.

Graphing Solutions on a Number Line

Visualizing the solution of a linear equation on a number line can help solidify understanding of where the solution lies relative to other numbers.
Here is how you can graph it:

• Draw a horizontal line, which will serve as your number line.
• Mark regular intervals, say numbers from -5 to 5, for clear context.
• Locate the solution on the line. For \(x = 1\), find number 1 on the line.
• Place a dot or a circle around the number 1, indicating the precise point where the solution sits.

Graphing on a number line provides a tangible way to see where the solution fits into the broader numerical picture and can be a helpful reference for more complex equations later.