Problem 78 - Algebra A Combined Function

Question

Use the addition property to fill in the blank so that the middle equation simplifies to the last equation. See the Concept Check in this section. $$ \begin{aligned} a+9 &=15 \\ a+9+(\quad) &=15+(\quad) \\ a &=6 \end{aligned} $$

Step-by-Step Solution

Answer

Add \(-9\) to both sides to simplify to \(a = 6\).

Step 1: Understanding the Problem

We need to determine what number should be added to both sides of the equation \(a + 9 = 15\) so that it simplifies to \(a = 6\). The goal is to isolate the variable \(a\).

Step 2: Identify Required Operation

To isolate \(a\), we should remove \(9\) from the left side of the equation \(a + 9\). To do this, we need to add \(-9\) to both sides of the equation.

Step 3: Apply the Addition Property

Add \(-9\) to both sides of the equation: \(a + 9 + (-9) = 15 + (-9)\). This simplifies to \(a = 6\) because the \(+9\) and \(-9\) cancel each other on the left side.

Key Concepts

Isolate the Variable

When solving equations, one of the main goals is to isolate the variable. In simpler terms, this means getting the variable by itself on one side of the equation. Consider the exercise: we start with the equation \( a + 9 = 15 \). The objective here is to isolate \( a \). To do this, we need to get rid of the number that is with \( a \) on the same side of the equation, which in our case is \( 9 \). Isolating the variable helps us find out the exact value of \( a \) by simplifying the equation into a direct statement where \( a = \ ext{some number} \). It’s like peeling back layers of an onion until what you’re looking for is all that's left.

Solving Linear Equations

Linear equations are foundational in mathematics and often appear in the form \( ax + b = c \). Solving these involves finding the value of the variable that makes the equation true. The exercise problem is a linear equation where you need to find \( a \).Steps in solving these equations typically include:

• Identify the operations being performed on the variable.
• Use inverse operations (like addition or subtraction) to undo these operations and get the variable on its own.
• Perform the same operation on both sides of the equation to maintain equality.

Once you isolate the variable, you have your solution, just like in our example which we simplified to \( a = 6 \). Practice is key here; the more you practice, the quicker and more intuitive it becomes to solve these equations.

Inverse Operations

Inverse operations are essential tools in solving equations, especially when isolating the variable. They are operations that undo each other. The simplest examples are addition and subtraction or multiplication and division.In the given example, to isolate \( a \) in the equation \( a + 9 = 15 \), we notice that \( 9 \) is being added to \( a \). To "undo" this addition, we use the inverse operation, which is subtraction. Thus, we add \(-9\) to remove the \(9\) next to \( a \). Subtraction in this step acts as the inverse operation for addition.Remember, whatever you do to one side of the equation, you must do to the other to keep the balance. This approach not only helps in isolating variables but is a fundamental principle when solving all kinds of equations.