Problem 5 - Algebra A Combined Function

Question

Twice the difference of a number and 8 is equal to three times the sum of the number and 3 . Find the number.

Step-by-Step Solution

Answer

The number is -25.

Step 1: Translate the problem into an equation

First, let's define the unknown number as a variable, let's say \(x\). The problem states: "Twice the difference of a number and 8" which translates to \(2(x - 8)\). Likewise, "three times the sum of the number and 3" translates to \(3(x + 3)\). Thus, the equation from the problem is \(2(x - 8) = 3(x + 3)\).

Step 2: Expand both sides of the equation

Now, we expand both the left and right sides of the equation. On the left side, \(2(x - 8)\) expands to \(2x - 16\). On the right side, \(3(x + 3)\) expands to \(3x + 9\). Thus, the equation is \(2x - 16 = 3x + 9\).

Step 3: Simplify the equation

We need to get all terms involving \(x\) on one side and constant terms on the other. Subtract \(2x\) from both sides to get: \(-16 = x + 9\).

Step 4: Solve for the unknown number

Subtract 9 from both sides to isolate \(x\). So, we have \(-16 - 9 = x\) which simplifies to \(x = -25\).

Key Concepts

Variable Definition

In mathematics, variables act like placeholders for unknown values. In our exercise, we're trying to find a mysterious number. Since its actual value isn’t given at the start, we use a symbol, most often a letter like \(x\), to represent it. This variable will stand in for our number until we solve the equation. This approach makes it easier to handle complex operations and solve for the number using equations. Remember, defining a variable is a critical first step in forming a mathematical equation.

Equation Translation

This part involves taking the words from the original problem and turning them into a mathematical equation. Here, we have the statement "Twice the difference of a number and 8," which converts to the expression \(2(x - 8)\). The phrase "three times the sum of the number and 3" translates to \(3(x + 3)\). By interpreting these phrases correctly, we accurately translate a word problem into a solvable mathematical expression. The full equation then becomes \(2(x - 8) = 3(x + 3)\). This critical step is like setting the stage for the drama of solving equations.

Equation Expansion

Now it's time to expand the equation. This is all about breaking down expressions like \(2(x - 8)\) and \(3(x + 3)\) to make them simpler. Expanding means multiplying each term inside the parenthesis by the number outside. The expression \(2(x - 8)\) becomes \(2 \times x - 2 \times 8 = 2x - 16\). On the other side, \(3(x + 3)\) becomes \(3 \times x + 3 \times 3 = 3x + 9\). Expanded equations are easier to work with, especially when it's time to solve for the variable.

Solving Equations

Solving equations is the grand finale where we find out what our variable stands for. Starting with the expanded equation \(2x - 16 = 3x + 9\), our goal is to isolate \(x\). By consolidating terms, we move all terms with \(x\) to one side by subtracting \(2x\) from both sides: \(-16 = x + 9\). Next, we get rid of numbers on the same side as \(x\) by subtracting \(9\) from each side of the equation, giving us \(-25 = x\). This results in \(x = -25\), revealing our mystery number. Remember, the solution offers the value for \(x\), the unknown number we initially defined.