Solving Equations with Multiple Variables

Algebra is the cornerstone of advanced mathematics, and understanding how to solve complex equations with multiple variables is essential for any student aiming to excel in the field. In this article, we will explore a method to solve a system of equations involving two variables and multiple digits within the equations. This approach will be demonstrated through a step-by-step example.

Problem Introduction

We will solve the following system of equations:

3a5b 26 a5b 22 2a 4 a 4/2 2 25b 22 5b 22 - 2 b 20/2 b 10

Step-by-Step Solution

We begin by solving the equations systematically. The equations we have are:

3a5b 26 i

a5b 22 ii

2a 4

a 4/2 2

25b 22

5b 22 - 2

b 20/2

b 10

Step 1: Solve for 'a' in the third equation:

}

2a 4 (1)

a 4/2 2

Step 2: Substitute the value of 'a' from step 1 into the first equation:

3a5b 26

3 * 25b 26

65b 26

5b 26 - 6

5b 20

b 20 / 5

b 4

Verification

We can also verify the value of 'b' by substituting 'a' into the second equation:

a5b 22

25b 22

5b 22 - 2

5b 20

b 20 / 5

b 4

Thus, we have found that a 2 and b 4.

Summary and Conclusion

In conclusion, solving a system of equations with multiple variables involves following a step-by-step approach. By simplifying the equations and substituting known values, we can accurately determine the values of the variables. This method is crucial for understanding and solving more complex mathematical problems in algebra and beyond.

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