Problem 15 - Fundamentals of Mathematics

Question

Solve each equation. $$ m+8=-1 $$

Step-by-Step Solution

Answer

The solution is \( m = -9 \).

Step 1: Understanding the Equation

The given equation is \( m + 8 = -1 \). This is a simple linear equation that we need to solve for the variable \( m \).

Step 2: Isolate the Variable

To find the value of \( m \), we need to isolate it on one side of the equation. We can do this by performing the operation of subtraction to both sides of the equation.

Step 3: Subtract 8 from Both Sides

Subtract 8 from both sides of the equation to isolate \( m \):\[ m + 8 - 8 = -1 - 8 \] This simplifies to:\[ m = -9 \]

Step 4: Solution Verification

To verify the solution, substitute \( m = -9 \) back into the original equation and check if both sides are equal: \[ (-9) + 8 = -1 \]Since both sides equal \(-1\), our solution is correct.

Key Concepts

Variable Isolation

Variable isolation is an important first step when solving equations. It means getting the variable alone on one side of the equation. This allows us to find its value easily. In the equation \( m + 8 = -1 \), our goal is to isolate \( m \). To do this, we need to remove \( 8 \) from the left side.

How can we achieve this? We perform the inverse operation of addition, which is subtraction. So, we subtract \( 8 \) from both sides of the equation. Why both sides? Because equations are like a balance scale; you must treat both sides equally to maintain balance.

• Start with: \( m + 8 = -1 \)
• Subtract \( 8 \) from both sides: \( m + 8 - 8 = -1 - 8 \)

This results in \( m = -9 \), with \( m \) isolated and ready for the next steps.

Equation Solving

Once we have isolated the variable, the process of solving a linear equation becomes more straightforward. The equation \( m = -9 \) means that we have already determined the value of \( m \) as \(-9\).

Linear equations, like the one in our exercise, are characterized by having the highest power of the variable as 1. They are usually simple and require basic arithmetic operations like addition, subtraction, multiplication, or division to isolate and solve for the variable.

• Identify the operation needed to isolate \( m \).
• Perform that operation equally on both sides.
• Simplify to find the variable's value.

Following these basic steps ensures consistent results in solving different equations.

Solution Verification

After solving any equation, checking your solution is crucial. This step is called solution verification. It ensures that the solution satisfies the original equation. To do this, you substitute the calculated value back into the original equation and see if the two sides balance.

For our solution where \( m = -9 \):

• Original equation: \( m + 8 = -1 \).
• Substitute \( m = -9 \): \((-9) + 8 = -1 \).

Both sides simplify to \(-1\), confirming our solution is accurate.

This step reassures that no mistakes were made during calculation and strengthens your mathematical confidence. Always verify your results when solving equations!