Problem 85 - Algebra 1

Question

Solve the equation. $$ 3(r-1)=2(r-2) $$

Step-by-Step Solution

Answer

The solution of the equation \(3(r-1) = 2(r-2)\) is \(r = -1\)

Step 1: Simplify on both sides

To get started, first simplification is essential. By using distributive property (which means a(b+c) = ab + ac), Simplify both sides of the equation. So, multiply \(3\) through the parentheses on the left side and \(2\) on the right side of the equal sign: \(3r - 3 = 2r - 4\)

Step 2: Rearrange the equation

Next step is to make one side of the equation a variable term and another side as numerical term. To do this, subtract \(2r\) from both sides and then add \(3\) to both sides of the equation: \(3r - 2r = -4 + 3\), so we get \(r = -1\)

Step 3: Check the solution

The final step is to check their solution to make sure it's correct. Substitute \(r = -1\) into the original equation and see if it verifies the equation: \(3(-1-1)=2(-1-2)\). If the left side of the equation equals to the right side, then the solution is correct, and indeed in this case, \(-6 = -6\), thus solution is confirmed

Key Concepts

Distributive Property

Understanding the distributive property is key to solving many algebraic equations. This property allows you to simplify expressions in a way that makes them easier to work with. The basic idea is that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results.
For example, consider the expression \(a(b+c)\). According to the distributive property, this can be rewritten as \(ab + ac\). This method helps in expanding expressions and simplifying equations.

• In our exercise, the distributive property helps in rewriting \(3(r-1)\) as \(3r - 3\), and \(2(r-2)\) as \(2r - 4\).
• This approach ensures that every term inside the parentheses is multiplied by the number outside, effectively removing the parentheses.

By using the distributive law, complex problems become simpler, and you're left with a more manageable expression to work with.

Simplification

Simplification is a vital step in solving equations as it makes complex problems more manageable. Once you've applied the distributive property, the next task is to combine like terms to further simplify.
In our equation, after distributing, we have \(3r - 3 = 2r - 4\). The goal is to isolate the variable on one side. This involves:

• Identifying like terms, which in this case are \(3r\) and \(2r\), and constants, which are \(-3\) and \(-4\).
• Rearranging the equation such that all terms containing the variable are on one side and constant terms on the other.

By subtracting \(2r\) from both sides and simplifying the constants, you get \(r = -1\). With fewer, simpler terms, solving the equation clearly becomes easier.
Simplification helps reduce errors and clarifies each step, ensuring that every operation leads you closer to the solution.

Checking Solutions

After finding a solution, it's crucial to verify its correctness. Checking solutions ensures you're on the right track and confirms that the process was done accurately.
To check the solution for an equation like \(3(r-1) = 2(r-2)\), substitute the found value, \(r = -1\), back into the original equation. The idea is to see if both sides of the equation are equal:

• Substitute \(-1\) where \(r\) appears in the equation and simplify each side.
• For this problem, replacing \(r\) with \(-1\) yields: \(3(-1-1)\) and \(2(-1-2)\).
• Simplify both sides: \(3(-2) = 2(-3)\) resulting in \(-6 = -6\).

If both sides of the equation equal each other, as they do here, your solution is correct. This step of checking reinforces your understanding and assures that no mistakes were made in earlier steps.