Problem 27 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$x+4-7=3-10$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -4 \).

Step 1: Simplify the Left Side

Begin by simplifying the left side of the equation. We have: \[ x + 4 - 7 \]Combine like terms (constants in this case), we get:\[ x - 3 \]

Step 2: Simplify the Right Side

Now, simplify the right side of the equation:\[ 3 - 10 \]Perform the subtraction to get:\[ -7 \]

Step 3: Rewrite the Simplified Equation

Rewrite the equation using the simplified expressions from both sides from Steps 1 and 2. This gives us:\[ x - 3 = -7 \]

Step 4: Apply the Addition Property of Equality

To isolate \( x \), we use the addition property of equality. Add 3 to both sides of the equation:\[ x - 3 + 3 = -7 + 3 \]This simplifies to:\[ x = -4 \]

Key Concepts

Simplification of Expressions

Simplifying expressions is all about making them easier to work with by reducing complexity. This means transforming the expression into a simpler or more straightforward form without changing its value. In our task, it involves:

• Combining numbers, known as constants, and similar variable terms.
• Performing arithmetic operations to simplify numerical expressions.

Let's take the example given in the original exercise, starting with the left side of the equation: \[ x + 4 - 7 \]To simplify this side, identify and combine like terms. Here, the like terms are the constants 4 and -7. When you perform the subtraction, \[ 4 - 7 = -3 \]So, the expression simplifies to:\[ x - 3 \]Similarly, simplifying the right side, \[ 3 - 10 \]The subtraction results in:\[ -7 \]After both sides are simplified, the equation is rewritten in its simplest terms. Understanding this process is crucial because it sets the stage for effectively solving the equation.

Addition Property of Equality

The addition property of equality is a fundamental principle in algebra. It states that you can add the same number to both sides of an equation, and the equation will remain true. It's a vital tool used to isolate variables and solve equations.In our example, after simplifying the expression, we were left with:\[ x - 3 = -7 \]To solve for \(x\), you need to get rid of the -3 that's with \(x\). According to the addition property of equality, we do this by adding 3 to both sides:\[ x - 3 + 3 = -7 + 3 \]The purpose here is to "cancel out" the -3 on the left side. This simplifies to:\[ x = -4 \]Utilizing the addition property in this manner helps to maintain balance in the equation, ensuring it remains correct while you work toward finding the value of the variable. Understanding this not only helps in solving equations but is also useful in verifying your work.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions and equations. Like terms are terms that have the same variable raised to the same power. To identify like terms:

• Look for terms with the same variables and exponents.
• Numbers without variables are constants and can also be combined.

In the original example, the left side of the equation had the terms \(x + 4 - 7\). Here: - \(x\) is a variable term.- \(+4\) and \(-7\) are constant terms.Constant terms can be combined:\[ 4 - 7 = -3 \]Thus, simplifying the expression to: \[ x - 3 \]In solving equations, combining like terms helps to reduce the equation to a simpler form, simplifying the process of solving for the variable. This step is foundational because it makes the expression manageable and sets up the path to finding a solution more straightforwardly. Mastering this concept is key to excelling in algebra.