Problem 7 - Algebra A Combined Function

Question

The product of twice a number and three is the same as the difference of five times the number and \(\frac{3}{4}\). Find the number.

Step-by-Step Solution

Answer

The number is \(-\frac{3}{4}\).

Step 1: Translate the Word Problem into an Equation

Read the problem carefully to identify key information. The product of twice a number (let's call this number "x") and 3 can be represented as \(2x \times 3\). This is stated to be equal to the difference between five times the number and \(\frac{3}{4}\). The equation can be written as: \(6x = 5x - \frac{3}{4}\).

Step 2: Simplify the Equation

Begin by simplifying the equation \(6x = 5x - \frac{3}{4}\) by subtracting \(5x\) from both sides: \[6x - 5x = 5x - 5x - \frac{3}{4}\]This simplifies to: \(x = -\frac{3}{4}\).

Step 3: Verify the Solution

Substitute \(x = -\frac{3}{4}\) back into the original equations to verify.For the left side of the equation (product of twice the number and three):\[2 \times -\frac{3}{4} \times 3 = -\frac{6}{4} \times 3 = -\frac{18}{4} = -\frac{9}{2}\]For the right side of the equation (difference of five times the number and \(\frac{3}{4}\)):\[5 \times -\frac{3}{4} - \frac{3}{4} = -\frac{15}{4} - \frac{3}{4} = -\frac{18}{4} = -\frac{9}{2}\]Since both sides equal \(-\frac{9}{2}\), the solution \(x = -\frac{3}{4}\) is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of the variable that satisfies the given equation. In this exercise, we started by having the equation \(6x = 5x - \frac{3}{4}\). To solve it, we needed to isolate the variable \(x\). This is done by moving all terms involving \(x\) to one side of the equation and constants to the other.
To start, we subtracted \(5x\) from both sides:

• On the left: \(6x - 5x\)
• On the right: \(5x - 5x - \frac{3}{4}\)

Simplifying the expression gives \(x = -\frac{3}{4}\). This tells us that \(x = -\frac{3}{4}\) is the potential solution to our problem.
Understanding how to manipulate an equation is essential. Always remember to perform the same operation on both sides of the equation to maintain balance.

Translating Word Problems into Equations

Translating word problems into equations is a crucial step in solving algebraic problems. It involves converting the words and phrases of a problem into mathematical symbols and expressions. Let's break down the original problem: we needed to find a number where twice that number times three equals the difference of five times the number and \(\frac{3}{4}\).
Here's how we did it:

• "Twice a number": let the number be \(x\), then \(2x\) represents twice the number.
• "Product of twice a number and three": expressed as \(2x \times 3\) or simply \(6x\).
• "Difference of five times the number and \(\frac{3}{4}\)": this is represented by \(5x - \frac{3}{4}\).

Putting these expressions together, we formed the equation \(6x = 5x - \frac{3}{4}\). This step is all about identifying the math operations described by the words and translating them correctly.

Verification of Solutions

Verification of solutions is a pivotal step to ensure that the solution you obtained is indeed correct. While solving the textbook problem, we found that \(x = -\frac{3}{4}\). Verification involves substituting this value back into the original problem to see if it holds true.
For our exercise, substituting \(x = -\frac{3}{4}\) into the first part (product of twice the number and three):

• Calculate: \(2 \times -\frac{3}{4} \times 3 = -\frac{9}{2}\)

Then, for the second part (difference of five times the number and \(\frac{3}{4}\)):

• Calculate: \(5 \times -\frac{3}{4} - \frac{3}{4} = -\frac{9}{2}\)

Since both calculations yield \(-\frac{9}{2}\), it confirms that our solution \(x = -\frac{3}{4}\) is valid. Verification ensures accuracy and boosts confidence in your solution-finding skills.