Question

the sum of twice a number and a larger number is 145. the difference between the numbers is 55. let \\(x\\) represent the smaller number and \\(y\\) represent the larger number. which equations represent the situation? check all that apply.

\\(x - y = 55\\)
\\(2(x + y) = 145\\)
\\(2x + y = 145\\)
\\(y - x = 55\\)
\\(y = x + 55\\)

Explanation:

Identify the variables

Using the Variable Identification knowledge point, we define:

• Let \(x\) represent the smaller number.
• Let \(y\) represent the larger number.

Translate the first sentence

Using the Algebraic Translation knowledge point:

• "Twice a number" refers to twice the smaller number, which is \(2x\).
• "The sum of twice a number and a larger number is 145" translates to:

$$2x + y = 145$$

Translate the second sentence

Using the System of Equations Translation knowledge point:

• "The difference between the numbers is 55" means the larger number minus the smaller number equals 55:

$$y - x = 55$$

• We can also rewrite this equation by adding \(x\) to both sides:

$$y = x + 55$$

Match with the given options

Let's check each option against our translated equations:

• \(x - y = 55\): Incorrect (this would mean the smaller minus the larger is positive).
• \(2(x + y) = 145\): Incorrect (this represents twice the sum of both numbers).
• \(2x + y = 145\): Correct (matches our first equation).
• \(y - x = 55\): Correct (matches our second equation).
• \(y = x + 55\): Correct (equivalent to our second equation).

Answer:

*

$$x - y = 55$$

*

$$2(x + y) = 145$$

• **

$$2x + y = 145$$

(Correct answer)**

• **

$$y - x = 55$$

(Correct answer)**

• **

$$y = x + 55$$

(Correct answer)**