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Question 2 Translate the following verbal statement into an algebraic equation and then solve: Use \( x \) for your variable. The sum of four times a number and seven is ten. Equation Check Answer

Ask by Mitchell Crawford. in the United States
Mar 16,2025

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Tutor-Verified Answer

Answer

\( x = \frac{3}{4} \)

Solution

**Step 1: Translate the Statement into an Equation** The verbal statement "The sum of four times a number and seven is ten" can be interpreted as: \[ 4x + 7 = 10 \] --- **Step 2: Solve the Equation** 1. **Subtract 7 from both sides:** \[ 4x + 7 - 7 = 10 - 7 \quad \Longrightarrow \quad 4x = 3 \] 2. **Divide both sides by 4:** \[ x = \frac{3}{4} \] --- **Step 3: Verify the Solution** Substitute \( x = \frac{3}{4} \) back into the original equation: \[ 4\left(\frac{3}{4}\right) + 7 = 3 + 7 = 10 \] Since the left-hand side equals the right-hand side, the solution is correct. --- **Final Answer:** \[ x = \frac{3}{4} \]

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The Deep Dive

To translate the verbal statement into an algebraic equation, we first identify the components. "Four times a number" can be written as \( 4x \), and "the sum of four times a number and seven" is \( 4x + 7 \). We know this sum equals ten, so we write the equation as: \[ 4x + 7 = 10 \] Next, we solve for \( x \): 1. Subtract 7 from both sides: \( 4x = 10 - 7 \) \( 4x = 3 \) 2. Divide both sides by 4: \( x = \frac{3}{4} \) Now, let's check the answer by substituting \( x \) back into the original statement: Calculate \( 4 \times \frac{3}{4} + 7 \): \( 3 + 7 = 10 \) The check confirms the solution is correct! To further appreciate the context of algebra, remember that ancient civilizations like the Babylonians were solving equations around 2000 BC, using methods quite different from today's! They laid the groundwork for all of modern algebra. Additionally, in real-world applications, algebra provides a useful framework for budgeting, where setting equations can help determine how much to save, how much to spend, or how to balance your income with expenses effortlessly. So next time you budget, remember, you’re doing algebra!

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