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)
Define the variable: Let x be the number.
Set up the equation: x − 6 = 5 ( x + 2 ) .
Solve for x : x = − 4 .
The number is − 4 .
Explanation
Problem Analysis Let's analyze the problem. We are given a word problem that can be translated into an algebraic equation. We need to find the number that satisfies the given condition. Let's denote the number by x . The problem states that the difference of the number and 6 (which is x − 6 ) is the same as 5 times the sum of the number and 2 (which is 5 ( x + 2 ) ). So, we can set up the equation x − 6 = 5 ( x + 2 ) and solve for x .
Expanding the equation Now, let's solve the equation x − 6 = 5 ( x + 2 ) . First, we distribute the 5 on the right side of the equation: x − 6 = 5 x + 10
Isolating x Next, we want to isolate x on one side of the equation. Let's subtract x from both sides: − 6 = 4 x + 10
Further isolating x Now, subtract 10 from both sides: − 16 = 4 x
Solving for x Finally, divide both sides by 4 to solve for x : x = − 4
Verification Therefore, the number is -4. We can check our answer by plugging it back into the original equation: − 4 − 6 = 5 ( − 4 + 2 ) − 10 = 5 ( − 2 ) − 10 = − 10 The equation holds true, so our answer is correct.
Final Answer The number is -4, which corresponds to option A.
Examples
Understanding how to translate word problems into algebraic equations is a fundamental skill in mathematics. For example, suppose you are managing a budget for a school event. You know that the cost of renting the venue is a fixed amount, and you also need to account for the cost per student attending. By setting up an equation similar to the one in this problem, you can determine the number of students you can afford to invite based on your budget constraints. This involves defining variables, translating the given information into an equation, and solving for the unknown variable, just like we did in this problem.
