Problem 15 - Prealgebra

Question

Solve each equation. $$a+9=-6$$

Step-by-Step Solution

Answer

The solution is \( a = -15 \).

Step 1: Understand the Equation

The equation given is \( a + 9 = -6 \). This is a one-variable linear equation that we need to solve for \( a \). Our goal is to isolate \( a \) on one side of the equation.

Step 2: Isolate the Variable

To isolate \( a \), we need to get rid of \( 9 \) from the left side of the equation. We can do this by subtracting \( 9 \) from both sides: \( a + 9 - 9 = -6 - 9 \).

Step 3: Simplify Both Sides

Simplify both sides of the equation: On the left: \( a + 0 = a \). On the right: \( -6 - 9 = -15 \). Thus, the equation simplifies to \( a = -15 \).

Step 4: Verify the Solution

Substitute \( a = -15 \) back into the original equation to verify it: \( -15 + 9 = -6 \). Since both sides of the equation are equal, \( a = -15 \) is indeed the correct solution.

Key Concepts

One-Variable Linear Equations

A one-variable linear equation is a type of algebraic equation where there is only one variable, or unknown, such as "a" in our example. The equation is linear because the variable is not raised to any power other than one. Linear equations are considered fundamental in algebra because they model direct relationships between two expressions.
These equations are written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable to be found. Solving these equations usually involves isolating the variable to determine its value, as shown in the exercise where we solve for \( a \) in \( a + 9 = -6 \).
Understanding the basic structure of one-variable linear equations helps in predicting the variable's behavior and solution approach, as similar steps are used across different problems.

Isolating the Variable

Isolating the variable is a crucial step in solving any one-variable linear equation. It involves manipulating the equation such that the unknown is alone on one side of the equation. This is done by performing inverse operations on both sides to eliminate any numbers attached to the variable.

• For addition, perform subtraction.
• For subtraction, perform addition.
• For multiplication, perform division.
• For division, perform multiplication.

In the original exercise \( a + 9 = -6 \), we needed to isolate \( a \). To do this, we subtracted 9 from both sides. This left us with \( a \) alone on one side and simplified the equation to \( a = -15 \), showing that \( a \) equals \(-15\). The goal of isolating the variable is to have direct knowledge of what the variable equals.

Equation Verification

Once you find a solution to a one-variable linear equation by isolating the variable, the final step is to verify that solution. Verification involves substituting the value back into the original equation to ensure the equation holds true.
This step is essential because it confirms the accuracy of your solution. For example, in our exercise, after finding that \( a = -15 \), we substitute it back into the original equation to check: \( -15 + 9 = -6 \).
Both sides are equal, confirming that our solution for \( a \) is correct. Verification not only confirms the solution but also builds confidence in the problem-solving process, ensuring no errors have been made during isolation.