To solve the system of equations, we start with the second equation . We can rearrange it to express in terms of :

Now, we substitute this expression for into the first equation, :

Expanding this, we get:

Combining like terms yields:

Now, solving for :

Substituting back into the expression for :

Thus, the solution to the system of equations is:

Now, let’s engage with some additional insights!

Throughout history, systems of linear equations played a crucial role in fields like commerce and engineering, going back to ancient civilizations. The methods for solving these problems, including substitution and elimination, have evolved through mathematicians like Gauss and Cramer, shaping our modern understanding of algebra.

In real-world applications, systems of equations are everywhere—from calculating budgets and managing resources to optimizing production in factories. For instance, if a company wants to maximize profits based on varying prices of products, they often use systems of equations to find that sweet spot. The possibilities are endless!