Solving Simultaneous Equations: y x - 1 and xy 3

In this article, we will explore the process of solving a system of two linear equations in two variables. Specifically, we will tackle the problem of solving the simultaneous equations:

Equations:

y x - 1 xy 3

Method: Substitution

The substitution method is a powerful tool in algebra. It involves expressing one variable in terms of another and then substituting this expression into the other equation to solve for the remaining variable.

Step 1: Express One Variable in Terms of the Other

Given the first equation:

y x - 1

We can directly express y in terms of x.

Step 2: Substitute and Simplify

Now, substitute this expression into the second equation:

xy 3

Substitute y x - 1 into the second equation:

x(x - 1) 3

Expand and simplify:

x^2 - x 3

Move all terms to one side:

x^2 - x - 3 0

This is a quadratic equation, but in this case, we can use substitution to find the value of x.

Step 3: Solve for x

From the original equation y x - 1, we know that:

x - y 1

Substitute into the second equation:

x - (x - 1) 3

Subtract x - 1 from both sides:

x - x 1 3

1 3 - x x

2x - 1 3

Adding 1 to both sides:

2x 4

Divide by 2:

x 2 / 2

x 1

Step 4: Solve for y

Now, substitute x 1 back into y x - 1:

y 1 - 1

y 2

Conclusion

Thus, the solution to the simultaneous equations is:

x 1 y 2

This can be written as the pair (1, 2).

Additional Examples

For additional clarity, let's consider a different version of the solution:

Version 1:

Substitute x - 1 for y in the second equation:

x - (x - 1) 3

2x - 1 3

2x 4

x 2

Substituting x 1 back into y x - 1:

y 1 - 1

y 2

Version 2:

Using the substitution method, we can also solve the second equation by expressing the system as:

y x - 1

y - x -1

Add the two equations:

xy 3

y - x -1

2y 2

y 2

Substitute y 2 back into y x - 1:

2 x - 1

x 3

Version 3:

Graphically, the intersection of the two lines: