91Ó°ÊÓ

When solving systems of linear equations, the substitution method is a powerful tool. This method involves replacing one variable with an equivalent expression from another equation. Here's how it works:

In the given problem, we have two equations:

1) \(x - 5y = 8\)
2) \(x = 6y\)

We start by noticing that Equation 2 already expresses \(x\) in terms of \(y\). This makes it easy to substitute \(x\) in Equation 1. By doing this, we can reduce the two-variable equation into a single-variable equation, which is simpler to solve.

So, we replace \(x\) in Equation 1 with \(6y\) from Equation 2, leading to the calculation \(6y - 5y = 8\) and simplifying to \(y = 8\). Then, using \(y\) value in Equation 2, we find that \(x = 48\).

This method is useful when one equation is easily solved for one variable. It reduces the complexity and provides a straightforward path to finding the solution.

Linear equations are algebraic expressions that describe a straight line when graphed on a coordinate plane. These equations typically take the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

In our example, we have two linear equations:

1) \(x - 5y = 8\)
2) \(x = 6y\)

Both of these equations can be plotted as lines on a graph. The solution to the system of equations is the point where these two lines intersect.

Linear equations are foundational in algebra and appear in various real-life situations, such as calculating distances, predicting trends, and optimizing resources. Understanding how to manipulate and solve them is a crucial skill in math.

In mathematics, solutions to equations are often represented as ordered pairs \((x, y)\). This notation indicates the exact point on a coordinate plane where the values of \(x\) and \(y\) satisfy both equations in a system.

For our solution:
\[ (x, y) = (48, 8) \]

This ordered pair tells us that when \(x = 48\) and \(y = 8\), both of the original equations hold true.

Ordered pairs are essential in graphing and analyzing data. They help us pinpoint exact locations and understand relationships between variables. This concept is not only key in solving algebraic equations but also in fields like physics, engineering, and economics, where precise quantitative relationships are crucial.