Tutorial on Solving a Pair of Simultaneous Equations Using the Substitution Method
Solving equations is a critical skill in mathematics, especially when dealing with simultaneous equations. In this tutorial, we will walk through the steps of solving a pair of simultaneous equations using the substitution method. The equations we will solve are: x - y 2 and 2x - y 7.

Step-by-Step Guide

1. Solve the first equation for x. The first equation is x - y 2. Adding y to both sides, we get:

x 2 y (Equation 1)

2. Substitute x 2 y into the second equation. The second equation is 2x - y 7. Substituting x 2 y into the second equation:

2(2 y) - y 7

Expanding and simplifying:

4 2y - y 7

4 y 7

y 7 - 4

y 3 (Equation 2)

3. Substitute the value of y back into the first equation to find x. Using y 3 in the first equation:

x 2 y

x 2 3

x 5 (Equation 3)

The solution to the system of equations is x 5 and y 3.

Verification

To verify the solution, let's substitute x 5 and y 3 back into the original equations:

1. First equation: x - y 2

5 - 3 2 (True)

2. Second equation: 2x - y 7

2(5) - 3 7

10 - 3 7 (True)

Alternative Methods

While the substitution method is straightforward, some prefer the elimination method. Here is an alternative approach:

1. From the first equation: x - y 2 (Equation 1)

2. Multiply both sides of Equation 1 by 2 to match the coefficient of x in the second equation:

2x - 2y 4 (Equation 4)

3. Subtract Equation 1 from Equation 4:

(2x - 2y) - (2x - y) 4 - 7

2x - 2y - 2x y -3

-y -3

y 3 (Equation 5)

4. Substitute y 3 into Equation 1:

x - 3 2

x 5 (Equation 6)

The solution is again x 5 and y 3.

Using Matrices to Solve Simultaneous Equations

For more complex systems, using matrices can be helpful. Here's how you can solve the same pair of equations using R software:

library(matlib) A - matrix(c(1, -1, 2, -1), nrow2, byrowTRUE) colnames(A) - c('1', '2') b - c(2, 7) showEqn(A, b) solve(A, b)

This code will output the solution x 5 and y 3, which matches our earlier results.

Conclusion

Understanding how to solve systems of linear equations is essential. The substitution and elimination methods, as well as using matrices, are valuable tools in your mathematical toolkit. The key is to practice regularly to ensure accuracy and speed.

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