Problem 77 - 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. $$ \begin{aligned} x-4 &=-9 \\ x-4+(\quad) &=-9+(\quad) \\ x &=-5 \end{aligned} $$

Step-by-Step Solution

Answer

+4 to both sides.

Step 1: Understanding the Goal

We need to use the addition property of equality to transform the equation \(x - 4 = -9\) into \(x = -5\). The aim is to determine what should be added to both sides of the equation \(x - 4 = -9\) to achieve the equation \(x = -5\).

Step 2: Analyzing the Final Equation

The final equation given is \(x = -5\). This implies that whatever we do to both sides of the equation \(x - 4 = -9\) should result in isolating \(x\) on the left side, giving \(x = -5\).

Step 3: Determine the Addition Required

In the equation \(x - 4 = -9\), if we add 4 to both sides, we aim to cancel out \(-4\) with \(+4\) on the left side. So, we need to add \(+4\) to both sides.

Step 4: Apply the Addition Property

We fill in the blanks with \(+4\) for both sides of the equation. The middle equation becomes: \(x - 4 + 4 = -9 + 4\).

Step 5: Simplifying Both Sides

Simplify the left side: \(x - 4 + 4 = x\). Simplify the right side: \(-9 + 4 = -5\). Thus, we have the simplified equation: \(x = -5\). This confirms the solution is correct.

Key Concepts

Solving Linear Equations

Linear equations allow us to find the unknown variable by setting up a balance between the two sides of the equation. In this exercise, the problem was simplified through a series of steps that led us from the equation \(x - 4 = -9\) to \(x = -5\). Each side of the equation is treated as the balance beam of a scale, and whatever operation you do to one side, you must do to the other.

To solve a linear equation effectively:

• Begin by identifying the operations affecting the variable (e.g., addition, subtraction, etc.).
• Apply inverse operations to shift these influences off the variable.
• Simplify incrementally until the variable stands alone on one side of the equation.

This systematic approach helps ensure clarity and accuracy in finding the value of the unknown variable.

Isolating Variables

Isolating a variable is a crucial step in solving equations. By removing other terms from around the variable, you reveal its value. In this example, the variable \(x\) initially shared the equation with a \(-4\). To solve for \(x\), we needed it to be by itself.

To isolate \(x\):

• Recognize the term attached to the variable, here it is \(-4\).
• Add the inverse of that term, so \(+4\) to both sides of the equation to remove it.
• Check each step by simplifying both sides, ensuring \(x\) is now isolated.

This method maintains the equation's balance using the addition property, leading to a correct and isolated \(x\).

Basic Algebraic Operations

Algebraic operations are fundamental processes used to manipulate equations and expressions. In the example provided, we utilized the addition property, an algebraic operation, to solve the equation.

Here's how basic algebraic operations work:

• **Addition Property of Equality:** Ensures adding the same number to both sides keeps the equation balanced.
• **Subtraction:** Counteracts addition, used to remove terms from one side.
• **Multiplication/Division:** Used for changing coefficients, keeping equations balanced by applying the same operation on both sides.

Understanding and applying these operations with precision is key to correctly simplifying and solving equations. They form the core of basic algebra and are indispensable in math solutions.