Solving the Equations: AB 76, A - B 38, and A ÷ B

In this article, we will solve a set of algebraic equations that involve the variables A and B. We are given three equations:

AB 76 A - B 38 A ÷ B ?

Step 1: Adding the equations AB and A - B

AB 76 A - B 38 AB A - B 76 38 2A 114 A 57

This simplification comes from the fact that the B terms cancel out when we add the two equations.

Solving for B

Now that we have the value of A, we can substitute it back into one of our original equations to find B. Let's use the first equation AB 76:

AB 76 57B 76 B 76 / 57 B 19

We can verify this by substituting A 57 and B 19 back into the second equation:

A - B 38 57 - 19 38

This is correct, confirming our solution.

We are also asked to find A ÷ B. With A 57 and B 19, we simply divide 57 by 19:

A ÷ B 57 ÷ 19 A ÷ B 3

Let's verify this using the third equation, A ÷ B ?:

A ÷ B 57 ÷ 19 3

There are other ways to solve this set of equations. Here are a couple of alternative methods:

We can also solve it directly by substituting the value of A from the simplified equation into one of the original equations. Let's use AB 76:

57B 76 B 76 ÷ 57 B 19

Once we have B, the rest follows naturally.

Another approach involves algebraic manipulation:

AB 76 A - B 38 Divide AB by (A - B): AB / (A - B) 76 / 38 2 / 1 This simplifies to 2A / 2B 3 / 1 A / B 3

This confirms that A ÷ B 3 using a different approach.

The solutions to the given set of equations are:

AB 76, which is satisfied with A 57 and B 19 A - B 38, which is also satisfied with A 57 and B 19 A ÷ B 3

Thus, the final answer is A ÷ B 3.

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