Problem 11 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$m+10=-2$$

Step-by-Step Solution

Answer

The solution is \(m = -12\).

Step 1: Move Constants to the Other Side

To isolate the variable \(m\), we need to move the constant \(10\) to the other side of the equation. This is done by subtracting \(10\) from both sides: \(m + 10 - 10 = -2 - 10\).

Step 2: Simplify the Equation

After subtracting \(10\) from both sides, the equation becomes \(m = -12\).

Step 3: Check the Solution

Substitute \(m = -12\) back into the original equation to check if the solution is correct: \(-12 + 10 = -2\). This simplifies to \(-2 = -2\), confirming that \(m = -12\) is indeed a valid solution.

Step 4: Graph Solution on a Number Line

Draw a number line and locate the point \(-12\). Since the solution is \(m = -12\), mark this point with a solid dot to show that \(-12\) is included in the solution set.

Key Concepts

Checking Solutions

When solving linear equations, it's crucial to verify that your solution is correct. This process is known as "checking solutions". You achieve this by replacing the variable in the original equation with the solution you found. In our exercise, we determined that \( m = -12 \). To check this, substitute \( -12 \) back into the equation:

• Start with the original equation: \( m + 10 = -2 \).
• Substitute \( m \) with \( -12 \): \( -12 + 10 = -2 \).
• Calculate the left side: \( -12 + 10 = -2 \).
• Compare both sides: \( -2 = -2 \).

The equation holds true, so \( m = -12 \) is indeed the correct solution. This step reinforces confidence in your solution and ensures no careless mistakes were made.

Number Line Graphing

Graphing a solution on a number line is a visual way to represent your findings. It helps in understanding where the solution lies in relation to other numbers. Here's how you perform number line graphing for our solution, \( m = -12 \):

• First, draw a horizontal line and mark it at equal intervals.
• Identify the position of zero on this line, then find \( -12 \) by counting segments to the left.
• Once you've located \( -12 \), mark this number with a solid dot.

The solid dot signifies that \( m = -12 \) is included in the solution set. This visual aid can help you see the solution distribution, and it's especially useful when dealing with inequalities.

Integer Operations

Integer operations are fundamental in solving linear equations, as they involve adding and subtracting whole numbers. Let's look at how these operations helped solve our equation:

• We began with \( m + 10 = -2 \). Our first task was to isolate \( m \), and to do this, we subtracted \( 10 \) from both sides. Subtraction is one of the primary operations on integers.
• Performing \( 10 - 10 = 0 \) on the left side helped us eliminate the constant next to \( m \).
• On the right side, \( -2 - 10 \) was simplified to \( -12 \) with basic subtraction.

In linear equations, often all you need are integer operations to isolate variables and find solutions. Mastering these operations can greatly ease the process of solving not just equations, but a wide range of mathematical problems.