Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$2 x+5=13$$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Isolate the term with x.

Subtract 5 from both sides of the equation. This gives us the new equation \(2x = 13 - 5\).

Step 2: Simplify the equation.

Perform the subtraction operation to simplify the equation. This gives us \(2x = 8\).

Step 3: Solve for x.

To find the value of \(x\), divide both sides of the equation by 2. Therefore, \(x = 8/2\).

Step 4: Final calculation

Calculate the value of \(x\). That is, \(x = 4\).

Step 5: Check the solution

Insert the value of \(x\) into the original equation to verify that it satisfies the equation. It should hold that \(2*4 + 5 = 13\).

Key Concepts

Addition Property of Equality

The addition property of equality plays a crucial role when solving linear equations. This principle states that you can add or subtract the same value from both sides of an equation without changing the equation's truth. Think of the equation as a perfectly balanced scale. Adding or subtracting the same weight from both sides keeps it balanced.
In our original exercise, we have the equation: \[ 2x + 5 = 13 \]To isolate the term containing \( x \), you need to remove the 5 added to \( 2x \). To do this, subtract 5 from both sides:\[ 2x + 5 - 5 = 13 - 5 \]This simplifies to:\[ 2x = 8 \]This step ensures that the equation remains balanced while getting closer to finding the value of \( x \). This property is a fundamental tool in algebra as it helps in moving terms from one side to the other without losing the equality.

Multiplication Property of Equality

The multiplication property of equality is another essential concept that helps solve equations. It states that you can multiply or divide both sides of an equation by the same non-zero number without affecting the equality.In the simplified equation from our example:\[ 2x = 8 \]The goal is to solve for \( x \) by getting it alone on one side of the equation. Here, \( 2x \) means 2 times \( x \), so to isolate \( x \), divide both sides of the equation by 2. This gives us:\[ \frac{2x}{2} = \frac{8}{2} \]After simplifying, you find:\[ x = 4 \]This step is crucial because it allows us to determine the exact value of \( x \) by keeping the equation balanced, just like adding or subtracting.

Isolation of Variable

The isolation of a variable is the process of manipulating an equation until the variable of interest is on its own on one side of the equation. In linear equations like our example, this is typically achieved using the addition and multiplication properties of equality.
Initially, we have:\[ 2x + 5 = 13 \]The first step involves using the addition property of equality to subtract 5 from both sides, leading to:\[ 2x = 8 \]Following this, employ the multiplication property of equality by dividing both sides by 2 to get:\[ x = 4 \]By isolating \( x \), we determine its precise value. This process is vital for solving algebraic equations as it helps us see the direct relationship of \( x \) to a number. It simplifies the equation down to a form where x can be easily understood and calculated.

Solution Verification

Solution verification is an important step to ensure the solution obtained satisfies the original equation. It's a way to confirm that no mistakes were made during the calculation process. Once you believe you have found the solution, you should plug it back into the original equation to verify.
For our example, once we determined that \( x = 4 \), we substitute \( x \) back into the original equation:\[ 2x + 5 = 13 \]Replace \( x \) with 4:\[ 2(4) + 5 = 13 \]After performing the multiplication and addition on the left side, we have:\[ 8 + 5 = 13 \]The equation holds true, meaning our solution \( x = 4 \) is correct. This step protects against errors, confirming the solution works in the context of the problem, reinforcing the importance of validation in problem-solving.