Problem 13 - Algebra Form and Function

Question

Solve the equations. $$ 2(r+5)-3=3(r-8)+20 $$

Step-by-Step Solution

Answer

Answer: \(r = 11\)

Step 1: Distribute

First, distribute the 2 and the 3 on both sides of the equation: $$ 2(r+5)-3=3(r-8)+20 $$ $$ 2r+10-3=3r-24+20 $$

Step 2: Combine like terms

Next, combine the like terms on both sides of the equation: $$ 2r+7=3r-4 $$

Step 3: Isolate the variable r

Now, we need to isolate \(r\) by moving all terms containing \(r\) to one side of the equation and all constant terms to the other side. To do this, subtract \(2r\) from both sides and add \(4\) to both sides: $$ 2r+7-2r=3r-4-2r $$ $$ 7=r-4 $$

Step 4: Solve for r

Finally, we can solve for \(r\) by adding \(4\) to both sides of the equation: $$ 7+4=r $$ $$ r=11 $$ Therefore, the solution to the equation is \(r = 11\).

Key Concepts

Distributive Property

The distributive property is a powerful tool in algebra that helps to simplify expressions and solve equations. It's used when you need to multiply a single term by two or more terms inside parentheses. In the original exercise, this property is applied to simplify both sides of the equation
\(2(r+5)-3=3(r-8)+20\).

Here's how it works: when distributing, you multiply the term outside the parentheses by each term inside the parentheses. For example:

• Distribute the \(2\) in \(2(r+5)\). This becomes \(2 \times r + 2 \times 5\), which simplifies to \(2r + 10\).
• Similarly, distribute the \(3\) in \(3(r-8)\). This becomes \(3 \times r + 3 \times (-8)\), simplifying to \(3r - 24\).

After distributing, your equation becomes \(2r + 10 - 3 = 3r - 24 + 20\). Now, the equation is much easier to work with, and we can move on to combine like terms.

Combining Like Terms

Combining like terms is the next step in simplifying algebraic expressions after applying the distributive property. This step involves going through the equation and bringing together all terms that share the same variable or are constants. In our equation from the previous step, we now have:

• On the left side: \(2r + 10 - 3\) simplifies as follows: Since \(10\) and \(-3\) are both numbers, they are 'like terms'. So, combining them gives you \(2r + 7\).
• On the right side: \(3r - 24 + 20\) becomes easier to handle as well. The constants \(-24\) and \(+20\) combine to give \(3r - 4\).

By combining like terms, the equation is further simplified into \(2r + 7 = 3r - 4\). With the equation now in its simplest form, we are poised to solve for our variable, \(r\).

Solving for a Variable

The final objective in solving an equation is to find the value of the unknown variable—in this case, \(r\). The simplified equation after the previous steps is \(2r + 7 = 3r - 4\). The goal is to isolate \(r\) on one side of the equation.

Here's how you can do it:

• First, eliminate the \(2r\) on the left by subtracting it from both sides. This gives \(7 = 3r - 2r - 4\).
• This simplifies to \(7 = r - 4\). Now, \(r\) is isolated with a constant on the other side.
• To solve for \(r\), simply add \(4\) to both sides of the equation. This yields \(7 + 4 = r\) and thus \(r = 11\).

Solving for \(r\) allows us to find its specific value, which is \(11\) in this case. Each step is crucial in ensuring the accuracy and simplicity of solving algebra equations.