Problem 48 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$13-3 r+2+6 r-2 r-1=3+2 \cdot 9$$

Step-by-Step Solution

Answer

The solution to the equation is \( r = 7 \).

Step 1: Simplify Both Sides

Combine like terms on both sides of the equation: \( 13 - 3r + 2 + 6r - 2r - 1 = 3 + 2\cdot9 \) simplify to \( r + 14 = 21 \).

Step 2: Perform Addition Property of Equality to Isolate \( r \)

To isolate the \( r \) term, subtract 14 from both sides of the equation. This will give: \( r = 21 - 14 \).

Step 3: Calculate the Value of \( r \)

Now, calculate the value of \( r \) from the equation in Step 2: \( r = 7 \).

Step 4: Check Proposed Solution

Substitute \( r = 7 \) into the original equation to verify if the equation holds true. If it does, then the solution for \( r \) is correct. If not, recheck the steps.

Key Concepts

Solving Algebraic Equations

When faced with an algebraic equation, the ultimate goal is to find the value of the unknown variable that makes the equation true. The process usually involves several stages including simplifying expressions, performing operations equally on both sides, and keeping the equation balanced. It's crucial to work methodically, ensuring that each step sets the stage for the next, ultimately leading to a solution.
To illustrate, consider the equation from our exercise: \(13-3r+2+6r-2r-1=3+2\cdot9\). The initial step involves simplifying both sides by combining like terms—those with the same variable or constant—followed by applying properties of equality, such as the addition property, to isolate and solve for the unknown variable. The path to the solution is paced by logic and the structural rules of algebraic operations.

Combining Like Terms

Combining like terms is a method of simplification in algebra. Like terms are terms that have exactly the same variable raised to the same power. Only the coefficients of like terms are different, and it is the coefficients that you combine.

• To combine like terms, sum up or subtract their coefficients.
• This simplification makes the equation more manageable.
• In the exercise, we combined terms with \(r\) and constants separately: \(-3r + 6r - 2r\) simplifies to \(r\), and \(13 + 2 - 1\) simplifies to \(14\).

Understanding how to combine like terms effectively is crucial for solving equations neatly and efficiently.

Isolating Variables

The step of isolating the variable is where we manipulate the equation to have the variable on one side and the constants on the other. This is often done through the addition or subtraction property of equality, which states that you can add or subtract the same number from both sides of the equation without changing its solution.

• In our exercise, we subtracted 14 from both sides to isolate \(r\): \(r + 14 - 14 = 21 - 14\).
• This simplification yields the isolated variable: \(r = 7\).

Isolating the variable is a pivotal step, as it unveils the solution to the equation, which is the value of the unknown variable.

Checking Solutions in Algebra

After solving an equation and obtaining a proposed solution, it's essential to verify its correctness. Checking the solution involves plugging the value of the variable back into the original equation and seeing if it results in a true statement.

• In our example, substituting \(r = 7\) back into the original equation checks if the left-hand side equals the right-hand side.
• If both sides are equal, the solution is confirmed to be correct.
• If not, it indicates there may have been an error in the process which needs revisiting.

This final step is a good practice to ensure the integrity of your solution and reinforce your understanding of how to solve algebraic equations.