Problem 13 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$a+10=-4$$

Step-by-Step Solution

Answer

The solution is \(a = -14\).

Step 1: Isolate the Variable

To solve the equation \(a+10=-4\), we need to isolate \(a\). Subtract \(10\) from both sides of the equation to achieve this:\[ a + 10 - 10 = -4 - 10 \]Simplifying, we get:\[ a = -14 \]

Step 2: Check the Solution

Plug the solution back into the original equation to ensure its accuracy. Substitute \(a=-14\) into the equation \(a+10=-4\):\[ -14 + 10 = -4 \]Simplifying the left side gives:\[ -4 = -4 \]Since both sides equal, the solution \(a = -14\) is correct.

Step 3: Graph on a Number Line

To graph \(a = -14\) on a number line, draw a horizontal line and mark common integers such as \(-16, -15, -14, -13, -12, \) and so on. Place a point exactly on \(-14\) to represent the solution.

Key Concepts

Checking Solutions

After finding a solution for an equation, it's crucial to check if it is indeed correct. This step confirms that our solution satisfies the original equation. To check a solution:

• Substitute the solution back into the original equation.
• Perform the arithmetic operations as indicated in the equation.
• If both sides of the equation are equal after substitution, then the solution is verified.

In the case of the equation \( a + 10 = -4 \), we found that \( a = -14 \). Plugging \( -14 \) into the equation gives \( -14 + 10 = -4 \). Simplifying, we see that both sides become \( -4 \), confirming the solution is correct.
Checking solutions is essential because it prevents potential errors and assures us that the work done is accurate and reliable.

Number Line Graphing

Graphing an equation on a number line is a visual way to represent solutions. This method provides an easier understanding of where a solution lies concerning other numbers. Here is how you can graph \( a = -14 \):

• Draw a horizontal line and evenly space markings to represent integers.
• Label these markings with numbers such as \(-16, -15, -14, -13, \) and beyond, depending on space.
• Place a distinct point, often a dot or a circle, at the coordinate representing the solution, which in this case is \(-14\).

Graphing helps in visualizing numbers in their context. It can also assist in understanding more complex math topics, such as inequalities and functions. A clearly drawn number line can highlight the position of solutions relative to other values.

Isolation of Variable

The method of isolation of the variable is fundamental in solving equations. It involves rearranging the equation so that the unknown variable is alone on one side of the equation, and all other terms are on the opposite side. The goal is to simplify the equation step by step. Here's how it works:

• Identify the term containing the variable you want to solve for. In \( a + 10 = -4 \), the variable term is \( a \).
• Perform the necessary arithmetic operations to both sides of the equation to get rid of other numbers near the variable. Here, subtract \( 10 \) from both sides to form \( a = -14 \).

This technique is not limited to addition or subtraction. Depending on the equation, one might need to use division, multiplication, or more advanced operations like logarithms or exponents. Mastering isolation of the variable is vital for solving algebraic equations of any form.