Question

given: 4c - d = f, d = c - e, f = d + c; prove: 3c = d

Explanation:

Step1: Substitute \( f \) in \( 4c - d = f \)

Given \( f = d + c \), substitute into \( 4c - d = f \), we get \( 4c - d = d + c \).

Step2: Simplify the equation

Add \( d \) to both sides: \( 4c = 2d + c \). Then subtract \( c \) from both sides: \( 3c = 2d \)? Wait, no, wait. Wait, maybe there's a typo in the given? Wait, the problem says "Prove: \( 3c = d \)". Wait, let's re - check. Given \( 4c - d = f \), \( d = c - e \) (maybe this is a red herring?), \( f = d + c \). So substitute \( f = d + c \) into \( 4c - d = f \): \( 4c - d = d + c \). Subtract \( c \) from both sides: \( 3c - d = d \). Then add \( d \) to both sides: \( 3c = 2d \)? No, that's not matching the prove. Wait, maybe the given is \( 4c - d = f \), \( d = c - e \) (maybe \( e \) is a typo, or maybe I misread). Wait, the user's problem: "Given: \( 4c - d = f \), \( d = c - e \), \( f = d + c \); Prove: \( 3c = d \)". Wait, let's do it again. \( 4c - d = f \) and \( f = d + c \), so \( 4c - d = d + c \). Subtract \( c \) from both sides: \( 3c - d = d \). Add \( d \) to both sides: \( 3c = 2d \). That's not \( 3c = d \). Maybe there's a typo in the problem, but assuming the problem is correct as given, maybe I made a mistake. Wait, maybe the given is \( 4c - d = f \), \( d = c \) (no, the user wrote \( d = c - e \)). Alternatively, maybe the prove is \( 3c = 2d \), but the user said \( 3c = d \). Anyway, following the steps:

From \( 4c - d = f \) and \( f = d + c \), equate them: \( 4c - d = d + c \).

Subtract \( c \) from both sides: \( 3c - d = d \).

Add \( d \) to both sides: \( 3c = 2d \). But the problem says to prove \( 3c = d \), so maybe there's a mistake in the problem statement. However, if we assume that maybe the given is \( 2c - d = f \) instead of \( 4c - d = f \), then:

\( 2c - d = f \), \( f = d + c \), so \( 2c - d = d + c \), subtract \( c \): \( c - d = d \), add \( d \): \( c = 2d \), no. Alternatively, maybe \( 4c - 2d = f \). Anyway, based on the given steps, if we proceed with the given equations:

Wait, the user's hand - written statements: "4c - d = f", "d = c - e", "f = d + c", "Prove: 3c = d".

So let's do the substitution again:

1. \( 4c - d = f \) (Given)
2. \( f = d + c \) (Given)
3. Substitute 2 into 1: \( 4c - d = d + c \) (Substitution Property)
4. Subtract \( c \) from both sides: \( 3c - d = d \) (Subtraction Property of Equality)
5. Add \( d \) to both sides: \( 3c = 2d \) (Addition Property of Equality)

But this is not \( 3c = d \). So there must be a typo in the problem. However, if we assume that the first equation is \( 2c - d = f \), then:

1. \( 2c - d = f \) (Given)
2. \( f = d + c \) (Given)
3. Substitute 2 into 1: \( 2c - d = d + c \)
4. Subtract \( c \): \( c - d = d \)
5. Add \( d \): \( c = 2d \), still not. Alternatively, if the first equation is \( 4c - 2d = f \):

1. \( 4c - 2d = f \)
2. \( f = d + c \)
3. Substitute: \( 4c - 2d = d + c \)
4. Subtract \( c \): \( 3c - 2d = d \)
5. Add \( 2d \): \( 3c = 3d \)
6. Divide by 3: \( c = d \), not \( 3c = d \).

Alternatively, maybe the given is \( 4c - d = f \), \( f = 2d + c \), then:

1. \( 4c - d = 2d + c \)
2. Subtract \( c \): \( 3c - d = 2d \)
3. Add \( d \): \( 3c = 3d \)
4. Divide by 3: \( c = d \), still not.

Given that there might be a typo, but following the steps as per the problem:

If we ignore the inconsistency and assume that the steps lead to \( 3c = d \), maybe the problem has a typo and the first equation is \( 2c - d = f \), but we'll proceed with the given equations.

Answer:

There seems to be a typo in the problem statement as the given equations lead to \( 3c = 2d \) instead of \( 3c = d \). However, if we follow the substitution and simplification steps:

From \( 4c - d = f \) and \( f = d + c \), we get \( 4c - d = d + c \), then \( 3c = 2d \). If we assume a typo and the intended prove is \( 3c = 2d \), then the proof is as above.