Algebraic Equations Explained: Determining B and C

This problem asks us to determine the relationship between variables B and C, given two algebraic equations:

• Equation 1: \(A + B = A + C\)
• Equation 2: \(AB = AC\)

We will analyze each equation separately to see what conclusions can be drawn about B and C.

Analyzing the First Equation: \(A + B = A + C\)

Let's start with the first given equation and simplify it to find the relationship between B and C.

Given: \[A + B = A + C\]

To simplify this equation, we can perform the same operation on both sides to maintain equality. The most straightforward step here is to subtract the variable A from both sides of the equation.

Subtract A from both sides: \[(A + B) - A = (A + C) - A\]

This simplifies to: \[B = C\]

From the first equation, we can directly conclude that B must be equal to C.

Solving the Second Equation: \(AB = AC\)

Now, let's examine the second equation provided:

Given: \[AB = AC\]

To solve this equation for B and C, we need to consider two possible cases for the variable A.

Case 1: When A is Not Equal to Zero (\(A \neq 0\))

If A is any non-zero number, we can divide both sides of the equation by A. Dividing by a non-zero number is a valid algebraic operation that maintains the equality.

Divide both sides by A: \[\frac{AB}{A} = \frac{AC}{A}\]

This simplifies to: \[B = C\]

So, if A is not zero, the second equation also leads to the conclusion that B is equal to C.

Case 2: When A is Equal to Zero (\(A = 0\))

We must consider the possibility that A could be zero. If A is zero, then division by A is undefined, so we cannot simply divide both sides by A.

Let's substitute \(A = 0\) into both original equations:

• For the first equation: \(A + B = A + C\)

  Substitute \(A = 0\):

  \[0 + B = 0 + C\] \[B = C\]

  This still implies that B must be equal to C.

• For the second equation: \(AB = AC\)

  Substitute \(A = 0\):

  \[0 \times B = 0 \times C\] \[0 = 0\]

  This statement \(0 = 0\) is always true, regardless of the values of B and C. However, since the first equation \(A+B=A+C\) (which also holds true when A=0) has already established that \(B=C\), this \(0=0\) result from the second equation does not contradict or change our finding. It simply means that if A is zero, the second equation provides no new information about B and C beyond what the first equation gives.

Conclusion on Variables B and C

From our analysis of both equations, we consistently arrive at the same conclusion:

• The first equation \(A + B = A + C\) directly simplifies to \(B = C\).
• The second equation \(AB = AC\) also simplifies to \(B = C\) when \(A \neq 0\).
• When \(A = 0\), the first equation still yields \(B = C\), and the second equation becomes \(0 = 0\), which is consistent with \(B = C\).

Therefore, given both conditions \(A + B = A + C\) and \(AB = AC\), the only logical conclusion is that B is equal to C.