Select the correct answer.
The difference of a number and 6 is the same as 5 times the sum of the number and 2. What is the number?
A. 4
B. -2
C. -1
D. 1
Answer (1)
Represent the unknown number with the variable x .
Translate the word problem into an algebraic equation: x − 6 = 5 ( x + 2 ) .
Solve the equation by expanding, simplifying, and isolating x .
The number is − 4 .
Explanation
Understanding the Problem Let's break down this word problem step by step to find the mystery number!
Translating to an Equation We're told that 'the difference of a number and 6' is the same as '5 times the sum of the number and 2'. Let's translate this into an equation. We'll use 'x' to represent the unknown number.
Setting up the Equation The difference of a number and 6 can be written as: x − 6 5 times the sum of the number and 2 can be written as: 5 ( x + 2 ) Since these two expressions are equal, we can set up the equation: x − 6 = 5 ( x + 2 )
Expanding the Equation Now, let's solve for x. First, we need to distribute the 5 on the right side of the equation: x − 6 = 5 x + 10
Isolating the x Term Next, we want to get all the x terms on one side of the equation. Let's subtract x from both sides: − 6 = 4 x + 10
Further Isolating x Now, let's isolate the term with x by subtracting 10 from both sides: − 16 = 4 x
Solving for x Finally, to solve for x, we divide both sides by 4: x = − 4
Finding the Number So, the number is -4.
Examples
Imagine you're trying to balance a seesaw. On one side, you have a weight that's 'a number minus 6'. On the other side, you have 5 times the weight of 'the number plus 2'. To find the exact number that makes the seesaw balance perfectly, you solve the equation we just worked through. This kind of problem-solving is useful in many real-life situations, like balancing budgets, calculating mixtures, or determining optimal quantities in manufacturing.
