Problem 26 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$59=s+90$$

Step-by-Step Solution

Answer

The solution is \( s = -31 \).

Step 1: Isolate the Variable

To isolate the variable \( s \), we need to get it by itself on one side of the equation. We can do this by subtracting 90 from both sides of the equation. Starting with the original equation: \[59 = s + 90\]Subtract 90 from both sides: \[59 - 90 = s + 90 - 90\]This simplifies to:\[-31 = s\]Now we have the variable isolated.

Step 2: Check the Solution

We substitute the value of \( s \) back into the original equation to verify our solution. Start with the original equation:\[59 = s + 90\]Substitute \( s = -31 \)\[59 = -31 + 90\]Calculate the right side:\[59 = 59\]Since both sides of the equation are equal, \( s = -31 \) is the correct solution.

Step 3: Graph the Solution on a Number Line

To graph \( s = -31 \) on a number line, draw a horizontal line and mark a point at \(-31\). Indicate the position of \( -31 \) with a filled circle or dot to show this is the exact solution. Other numbers should be marked for reference, such as -30 and -32, to give context.

Key Concepts

Isolating Variables

Isolating the variable is a fundamental skill in solving equations. The primary goal is to get the variable by itself on one side of the equation. In the given problem, we have the equation \(59 = s + 90\). Here, the variable to isolate is \(s\). To do this, we need to remove any numbers or operations attached to it.

• First, identify the operation linked with the variable. In this case, it is addition (\(+90\)).
• Next, perform the inverse operation to both sides of the equation. Since the term \(+90\) is added to \(s\), we subtract \(90\) from both sides.
• This yields \(59 - 90 = s\), which simplifies to \(-31 = s\). The variable \(s\) is now isolated.

The result \(-31 = s\) tells us that \(s\) equals \(-31\). This straightforward process allows us to tackle more complex equations by applying similar steps to isolate variables effectively.

Checking Solutions

Ensuring that the solution to an equation is correct is crucial. This step is called "checking solutions" and involves substituting the result back into the original equation.

• Start by returning to the original equation, \(59 = s + 90\).
• Substitute the isolated value, \(s = -31\), into the equation: \(59 = -31 + 90\).
• Perform the arithmetic on the right side of the equation: \(-31 + 90 = 59\).
• Since the left side (\(59\)) and the right side (\(59\)) are equal, the solution \(s = -31\) is verified as correct.

Checking solutions helps prevent errors and ensures that the steps taken to solve the equation were correct. It also confirms that the isolated value meets the requirements of the original equation.

Graphing on Number Line

Graphing solutions on a number line provides a visual representation of the solution. It shows the accurate position of a solution within the context of other numbers.

• Begin by drawing a horizontal line to represent the number line.
• Mark several key points on the line, such as \(-32, -31, -30\), for reference.
• Locate the number \(-31\) on this number line.
• Draw a filled circle or dot at \(-31\) to indicate the exact solution of the equation \(s = -31\).

Graphing on a number line is a simple yet effective way to visually confirm that the value of \(s\) places correctly within the number spectrum. It also aids students in understanding how equations and their solutions fit into broader numerical contexts.