Amy and Aaron each have a card with a number written on it. Amy's number is $x$ and Aaron's number is $y$.
(a) If Amy multiplies her number by 6 and subtracts it by 10, the result is the same as 2 times Aaron's number.
Show that $3x - y = 5$
Answer (2)
Translate the word problem into an equation: 6 x − 10 = 2 y .
Divide both sides of the equation by 2: 3 x − 5 = y .
Rearrange the equation to isolate 3 x − y : 3 x − y = 5 .
Conclude that 3 x − y = 5 is shown. 3 x − y = 5
Explanation
Understanding the Problem Let's break down the problem. We are given that Amy's number is x and Aaron's number is y . We also know that if Amy multiplies her number by 6 and subtracts 10, the result is the same as 2 times Aaron's number. Our goal is to show that 3 x − y = 5 .
Forming the Equation First, let's translate the given information into an equation. 'Amy multiplies her number by 6 and subtracts 10' can be written as 6 x − 10 . '2 times Aaron's number' is 2 y . Since these two are equal, we have the equation: 6 x − 10 = 2 y
Simplifying the Equation Now, let's simplify the equation. We can divide both sides of the equation by 2: 2 6 x − 10 = 2 2 y 3 x − 5 = y
Rearranging the Equation Finally, we need to rearrange the equation to match the expression we want to show, which is 3 x − y = 5 . To do this, we can add 5 to both sides and subtract y from both sides: 3 x − 5 + 5 − y = y + 5 − y 3 x − y = 5
Conclusion Therefore, we have shown that 3 x − y = 5 .
Examples
Imagine you're at a carnival. Amy has x tickets, and Aaron has y tickets. If you know that 6 times Amy's tickets minus 10 equals twice the number of Aaron's tickets, this problem helps you find a direct relationship between their tickets. Knowing how to manipulate such equations is useful in many real-life scenarios, such as calculating costs, comparing quantities, or even managing resources.
We have demonstrated that the equation 3 x − y = 5 holds true by translating the problem into the equation 6 x − 10 = 2 y . After simplifying and rearranging, we confirmed the required expression. Therefore, the solution has been validated step-by-step.
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