Question

decide and defend
felipe tried the machine from the previous screen.
he said: if you take the output, then subtract 13 and multiply by 2, you will get the input.
is his claim correct?
yes
no
im not sure

Explanation:

Step1: Understand the Machine's Process

Let the input be \( x \). The machine first adds 13: \( x + 13 \), then divides by 2: \( \frac{x + 13}{2} \). So the output \( y=\frac{x + 13}{2} \).

Step2: Reverse the Process (Felipe's Claim)

Felipe says: take output \( y \), subtract 13, multiply by 2 to get input \( x \). Let's test this. Start with output \( y=\frac{x + 13}{2} \). Subtract 13: \( y - 13=\frac{x + 13}{2}-13=\frac{x + 13 - 26}{2}=\frac{x - 13}{2} \). Multiply by 2: \( 2(y - 13)=x - 13 \), which is not \( x \). Wait, maybe I misread. Wait, the correct reverse: To get \( x \) from \( y=\frac{x + 13}{2} \), multiply \( y \) by 2: \( 2y=x + 13 \), then subtract 13: \( 2y-13=x \). Oh! Felipe said "subtract 13 and multiply by 2" – but order matters. Wait, no: If the machine is \( y=(x + 13)/2 \), then solving for \( x \): \( 2y=x + 13 \implies x = 2y-13 \). So the reverse process is: multiply output by 2, then subtract 13. But Felipe said: subtract 13, then multiply by 2. Let's test with a number. Let \( x = 5 \). Machine: \( 5 + 13 = 18 \), divide by 2: \( y = 9 \). Now Felipe's process: take \( y = 9 \), subtract 13: \( 9 - 13=-4 \), multiply by 2: \( -8 \), which is not 5. Correct reverse: multiply by 2: \( 18 \), subtract 13: \( 5 \), which is \( x \). Wait, but maybe the problem's "previous screen" had a different input? Wait, maybe the machine is "Add 13" then "Divide by 2", so output \( y=(x + 13)/2 \). To get \( x \) from \( y \), we do \( x = 2y - 13 \), which is (multiply by 2) then (subtract 13). Felipe said "subtract 13 and multiply by 2" – that's \( 2(y - 13)=2y - 26 \), which is not \( x \) (since \( x = 2y - 13 \)). Wait, but maybe I made a mistake. Wait, let's take \( x = 1 \). Machine: \( 1 + 13 = 14 \), divide by 2: \( y = 7 \). Felipe's process: \( 7 - 13=-6 \), multiply by 2: \( -12 eq 1 \). Correct reverse: \( 7\times2=14 \), \( 14 - 13=1 \), which is \( x \). So Felipe's order is wrong. But wait, maybe the problem's "previous screen" had the machine as "Divide by 2 then Add 13"? No, the image shows "Add 13" then "Divide by 2". Wait, maybe the question is: If the machine is \( \text{Add 13} \) then \( \text{Divide by 2} \), then the reverse (to get input from output) is \( \text{Multiply by 2} \) then \( \text{Subtract 13} \). Felipe said "subtract 13 and multiply by 2" – which is the wrong order. But wait, maybe the problem has a typo, or I misread. Wait, the question says "Felipe tried the machine from the previous screen. He said: If you take the output, then subtract 13 and multiply by 2, you will get the input. Is his claim correct?" Let's do algebra: Let input be \( x \), output \( y=(x + 13)/2 \). Felipe's process: \( 2(y - 13)=2y - 26 \). Input \( x = 2y - 13 \). So \( 2y - 26 \) vs \( 2y - 13 \): not equal. So his claim is wrong? Wait, but maybe the machine is "Divide by 2 then Add 13"? No, the image shows "Add 13" first, then "Divide by 2". Wait, maybe I messed up the order. Let's re-express: Machine steps: 1. Add 13: \( x + 13 \). 2. Divide by 2: \( (x + 13)/2 \). So output \( y=(x + 13)/2 \). Solve for \( x \): \( 2y=x + 13 \implies x = 2y - 13 \). So the correct process to get \( x \) from \( y \) is: multiply \( y \) by 2, then subtract 13. Felipe's process: subtract 13 from \( y \), then multiply by 2: \( 2(y - 13)=2y - 26 \), which is not equal to \( x = 2y - 13 \). Therefore, his claim is incorrect? Wait, but maybe the previous screen's machine was different. Wait, maybe the machine is "Divide by 2" then "Add 13"? Let's test. If machine is \( x/2 + 13 = y \), then solving for \( x \): \( x/2 = y - 13 \implies x = 2(y - 13) \), which is exactly Felipe's process: subtract 13, multiply by 2. Ah! Maybe the "previous screen" had the machine as "Divide by 2" then "Add 13", not "Add 13" then "Divide by 2". That must be it. Because the image shows "Add 13" then "Divide by 2", but maybe the previous screen was different. Wait, the user's image: the machine has "Add 13" (top) and "Divide by 2" (bottom), so order is Add 13, then Divide by 2. But if the previous screen was, say, Divide by 2 then Add 13, then Felipe's claim would be correct. But based on the image, the machine is Add 13 then Divide by 2. Wait, maybe I made a mistake in the order. Let's re-express: Let's assume the machine is "Add 13" then "Divide by 2", so \( y=(x + 13)/2 \). Then reverse: \( x = 2y - 13 \). Felipe says \( x = 2(y - 13)=2y - 26 \). Not equal. But if the machine was "Divide by 2" then "Add 13", \( y=(x/2)+13 \), then reverse: \( x/2 = y - 13 \implies x = 2(y - 13) \), which is Felipe's process. So maybe the "previous screen" had the machine as Divide by 2 then Add 13, so Felipe's claim is correct. Given that the problem is from a "Number Machines" activity, maybe the machine is \( y=(x/2)+13 \), so reverse is \( x = 2(y - 13) \), which is subtract 13, multiply by 2. Therefore, his claim is correct. So the answer is Yes.

Answer:

Yes