Problem 50 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$4 r-3=5+3 r$$

Step-by-Step Solution

Answer

The solution to the equation is \( r = 8 \).

Step 1: Analyze the equation

The given equation is \(4 r-3=5+3 r\). Notice that there are \( r \) terms on both sides of the equation. To solve for \( r \), isolate the \( r \) terms on one side of the equation and the constant terms on the other side.

Step 2: Use the addition property of equality

Subtract \(3 r\) from both sides of the equation in order to get the \( r \) terms on one side: \(4 r - 3 r = 5 + 3 r - 3 r\), which simplifies to \(r - 3 = 5\).

Step 3: Solve for r

Next, add 3 to both sides of the equation to isolate \( r \): \(r - 3 + 3 = 5 + 3\). This gives \(r = 8\).

Step 4: Check the solution

Substitute \( r = 8 \) into the original equation and make sure that both sides of the equation are equal: \( 4 * 8 - 3 = 5 + 3 * 8 \), which simplifies to 29 = 29. Since both sides are equal, \( r = 8 \) is indeed the correct solution.

Key Concepts

Solving Equations

To solve equations with an unknown variable, you need to balance both sides of the equation. Think of it like a balanced scale. Whatever you do to one side, you must do to the other. This keeps the equality true.

Equations are statements that claim two mathematical expressions are equal. In our example, we have the equation:

• \(4r - 3 = 5 + 3r\)

The goal is to find the value of \(r\) that makes this statement true. By applying different properties of equality rules, like the addition property, you systematically rearrange the equation until \(r\) is isolated on one side. This helps you to clearly see the value of \(r\). This is often done in steps to simplify both sides until the variable is by itself. Remember, patience and attention to each step are key.

Isolating Variable

Isolating the variable is crucial when solving equations. It means getting your variable alone on one side of the equation.

Let's look at the original equation:

• \(4r - 3 = 5 + 3r\)

We need to move all \(r\) terms to one side of the equation. Using the addition property of equality, we subtract \(3r\) from each side. This gives:

• \(4r - 3r - 3 = 5 + 3r - 3r\)
• simplifying to \(r - 3 = 5\)

The next move is to eliminate the constant term on the side with \(r\). By adding 3 to both sides, we perform another step of isolation:

• \(r - 3 + 3 = 5 + 3\)
• resulting in \(r = 8\)

Once the variable is isolated, you've effectively solved the equation.

Checking Solutions

After solving an equation, it's crucial to verify that your solution is correct.

This ensures that no mistakes were made during the solving process. Simply substitute the solution back into the original equation and confirm that both sides are equivalent. Let's verify our solution for \(r = 8\):

• Starting with \(4 \times 8 - 3\)
• on the left side, we calculate \(32 - 3 = 29\)
• on the right side, calculate \(5 + 3 \times 8\)
• which simplifies to \(5 + 24 = 29\)

Since both sides are equal (29 = 29), our solution \(r = 8\) is correct. This step helps to catch any potential errors and verify the process to ensure accuracy.