Problem 14 - Prealgebra

Question

Solve each equation. Check your solution. $$3(a-3)=2(a+4)$$

Step-by-Step Solution

Answer

The solution is \(a = 17\).

Step 1: Distribute on both sides

First, distribute the numbers outside the parentheses into the terms inside the parentheses. For the left side, distribute 3: \[ 3(a-3) = 3 \cdot a - 3 \cdot 3 = 3a - 9 \] For the right side, distribute 2: \[ 2(a+4) = 2 \cdot a + 2 \cdot 4 = 2a + 8 \] So, the equation simplifies to: \[ 3a - 9 = 2a + 8 \]

Step 2: Isolate the variable terms

Next, we want to get all terms involving the variable \(a\) on one side of the equation. Subtract \(2a\) from both sides to isolate \(a\): \[ 3a - 2a - 9 = 8 \] This simplifies to: \[ a - 9 = 8 \]

Step 3: Solve for the variable

Now we'll solve for \(a\) by isolating it. Add 9 to both sides to do this: \[ a - 9 + 9 = 8 + 9 \] Which simplifies to: \[ a = 17 \]

Step 4: Check the solution

To ensure that our solution is correct, substitute \(a = 17\) back into the original equation: \[ 3(a-3) = 2(a+4) \] Substitute \(a\) with 17: \[ 3(17-3) = 2(17+4) \] Simplify inside the parentheses: \[ 3(14) = 2(21) \] Calculate both sides: \[ 42 = 42 \] Since both sides of the original equation are equal when \(a = 17\), our solution is confirmed.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that helps simplify expressions and solve equations quickly. It allows us to multiply a number by a group of numbers inside parentheses. This property states that multiplying a single term with terms inside the parentheses is the same as doing each multiplication separately, then adding or subtracting the results.
For example, the distributive property works like this:

• For the equation \(3(a - 3)\), you would distribute \(3\) to both \(a\) and \(-3\), giving us \(3 \cdot a - 3 \cdot 3\).
• Similarly for \(2(a + 4)\), distribute \(2\) into both \(a\) and \(4\), resulting in \(2 \cdot a + 2 \cdot 4\).

Breaking it down this way helps simplify the equation, making it easier to work with. By calculating these multiplications first, as in our original problem, the equation \(3a - 9 = 2a + 8\) surfaces, readying you for the next steps of solving.

Variables

Variables are symbols used in algebra to represent unknown values or quantities that can change. They serve as placeholders for whatever number might satisfy an equation or expression. In our example, the letter \(a\) is a variable that represents an unknown number we need to find.

Variables are crucial because:

• They allow us to write general rules or formulas, such as expressing the problem \(3(a-3) = 2(a+4)\) which can apply to any possible number \(a\).
• They make it easier to manipulate equations and solve for unknowns by performing operations like addition, subtraction, and distribution around these symbols.

In our case, after using the distributive property, our goal was to get the variable \(a\) by itself to find out what it specifically represents — which, in this scenario, turned out to be 17.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. The steps taken can vary based on the equation's complexity, but they often follow a structured approach to isolate the variable. Here's a quick overview of how to solve equations like \(3(a-3)=2(a+4)\):

• Step 1: Simplify Both Sides - Use the distributive property to remove the parentheses and simplify each side.
• Step 2: Isolate Variable Terms - Adjust the equation to place all terms with the variable on one side, subtract or add terms as needed. In our equation, \(3a - 9 = 2a + 8\) leads to \(a - 9 = 8\) after subtracting \(2a\) from both sides.
• Step 3: Solve for the Variable - Further manipulate the equation to solve for the unknown. For \(a - 9 = 8\), adding 9 to both sides isolates \(a\), resulting in \(a = 17\).
• Step 4: Check Your Solution - Substitute your solution back into the original equation to verify it satisfies the equality. This confirms that \(a = 17\) is correct because it produces \(42 = 42\) on both sides.

Solving equations is like finding a balance and ensuring both sides equate. This structured approach makes uncovering unknown values straightforward.