91Ó°ÊÓ

A system of equations is a collection of two or more equations with a common set of variables. In this exercise, we are dealing with two equations with two unknowns, namely \(a\) and \(b\).

• The goal is to find values for \(a\) and \(b\) that make both equations true at the same time.
• These types of problems are common in algebra and are usually solved by methods such as substitution, elimination, or graphing.

In this exercise, we chose the substitution method as it is often straightforward when one equation can easily express a variable in terms of the other. This was evident from the first equation \(2b - a = -1\), where we expressed \(a\) in terms of \(b\). By doing so, we reduced the system to a single equation with one variable, making it easier to solve.Systems can have various outcomes:

• One unique solution: both equations intersect at a single point.
• No solution: the lines are parallel and do not intersect.
• Infinite solutions: the lines overlap completely.

Linear equations are algebraic expressions representing straight lines when graphed. The general form of a linear equation in two variables is \(ax + by = c\). In this exercise, both equations are linear, meaning each represents a straight line.

• In the given system, the first equation is \(2b - a = -1\) and the second is \(3a + 10b = -1\).
• The coefficients (numbers multiplying the variables) define the slope and position of each line on a graph.

When solving these equations, rearranging terms is crucial to isolate a variable, as seen with \(a = 2b + 1\). This expression allows us to substitute and eventually solve for both \(a\) and \(b\) efficiently.The beauty of linear equations lies in their predictability and simplicity. Once a solution is found, it can be substituted back into both equations to verify its accuracy, proving their consistency and logical structure.

Verification is an essential step in solving systems of equations, ensuring that the found solutions satisfy all original equations.

• It's crucial because errors can occur during algebraic manipulation, and verifying helps catch these mistakes.

In this exercise, once we found \(b = -\frac{1}{4}\) and \(a = \frac{1}{2}\), we plugged these values back into the original equations:- **First Equation**: \(2(-\frac{1}{4}) - \frac{1}{2}\) simplifies to \(-1\), confirming correctness.- **Second Equation**: \(3(\frac{1}{2}) + 10(-\frac{1}{4})\) also simplifies to \(-1\), again confirming correctness.This process confirms that these values are indeed solutions of the original system. Successful verification gives confidence in the correctness of the solution and the methods used to achieve it. This step is so important that it's often considered part of the solution process itself, rather than just a final check. While it might seem unnecessary if you're sure about your answer, it never hurts to double-check—plus, when learning, it's a great way to reinforce your understanding of the methods used.