Select the correct answer. The sum of 2 times a number and 10 is the same as 6 times the difference of the number and 1. What is the number?
Answer (2)
Set up the equation: 2 x + 10 = 6 ( x − 1 ) .
Expand the equation: 2 x + 10 = 6 x − 6 .
Simplify and solve for x : 10 = 4 x − 6 ⇒ 16 = 4 x ⇒ x = 4 .
The number is 4 .
Explanation
Problem Analysis Let's analyze the problem. We are given a word problem that translates 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 sum of 2 times the number and 10 is the same as 6 times the difference of the number and 1. This can be written as the equation 2 x + 10 = 6 ( x − 1 ) .
Expanding the Equation Now, let's solve the equation for x . First, we expand the right side of the equation: 2 x + 10 = 6 x − 6
Isolating x Next, we want to isolate x on one side of the equation. We can subtract 2 x from both sides: 10 = 4 x − 6
Further Isolating x Now, add 6 to both sides: 16 = 4 x
Solving for x Finally, divide both sides by 4 to solve for x :
x = 4 16 = 4
The Answer Therefore, the number is 4.
Examples
This type of problem is commonly used in everyday situations where you need to compare two different scenarios and find the value that makes them equal. For example, imagine you are comparing two different cell phone plans. Plan A costs $10 per month plus $2 for every gigabyte of data you use. Plan B costs $6 per month, but you pay $6 for every gigabyte of data you use beyond a certain limit. To find out at what data usage the two plans cost the same, you would set up an equation similar to the one we solved. In this case, the equation would be 10 + 2 x = 6 + 6 x , where x is the number of gigabytes of data used. Solving this equation would tell you the data usage at which both plans cost the same.
The number that satisfies the equation is 4. We found it by setting up an equation based on the problem's conditions and solving for the variable. The final answer is 4 .
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