The sum of twice a number and a larger number is 145. The difference between the numbers is 55. Let [tex]$x$[/tex] represent the smaller number and [tex]$y$[/tex] represent the larger number. Which equations represent the situation? Check all that apply.
[tex]$x-y=55$[/tex]
[tex]$2(x+y)=145$[/tex]
[tex]$2x+y=145$[/tex]
[tex]$y-x=55$[/tex]
[tex]$y=x+55$[/tex]
Answer (2)
The equations that represent the situation are 2 x + y = 145 , y − x = 55 , and y = x + 55 . These equations correspond to the conditions given in the problem about the smaller and larger number. The other options do not correctly reflect the statements provided.
;
Translate the first statement into an equation: 2 x + y = 145 .
Translate the second statement into an equation: y − x = 55 .
Rewrite the second equation: y = x + 55 .
The equations that represent the situation are: 2 x + y = 145 , y − x = 55 , and y = x + 55 .
Explanation
Problem Analysis Let x represent the smaller number and y represent the larger number. We are given two pieces of information:
The sum of twice the smaller number and the larger number is 145.
The difference between the larger number and the smaller number is 55.
Translating the First Statement Let's translate the first statement into an equation. "The sum of twice the smaller number and the larger number is 145" can be written as:
2 x + y = 145
Translating the Second Statement Now, let's translate the second statement into an equation. "The difference between the larger number and the smaller number is 55" can be written as:
y − x = 55
Rewriting the Second Equation We can also rewrite the second equation as:
y = x + 55
Comparing with Given Options Now, let's compare the equations we derived with the given options:
x − y = 55 - Incorrect, as we have y − x = 55 .
2 ( x + y ) = 145 - Incorrect, as we have 2 x + y = 145 .
2 x + y = 145 - Correct.
y − x = 55 - Correct.
y = x + 55 - Correct.
Final Answer Therefore, the equations that represent the situation are:
2 x + y = 145
y − x = 55
y = x + 55
Examples
Understanding how to translate word problems into algebraic equations is a fundamental skill in mathematics with numerous real-world applications. For instance, consider a scenario where you're managing a budget for a school event. You know that the total cost for renting the venue and buying refreshments cannot exceed a certain amount. By expressing the costs as variables and setting up equations based on the given constraints, you can determine the maximum number of attendees you can accommodate or optimize your spending to stay within budget. This process of translating real-world scenarios into mathematical models allows for informed decision-making and efficient problem-solving.
