91Ó°ÊÓ

An ordered pair consists of two elements presented in a particular order, usually written as (x, y). These are used to specify a location in a two-dimensional plane called the Cartesian coordinate system. In the context of solving equations, each ordered pair is a potential solution that satisfies the given equation when values for x and y are substituted. The coordinates represent locations that can be plotted on a graph.

For linear equations like the one in our task, finding ordered pairs involves choosing a value for x and then calculating the corresponding y. This results in pairs like (-15, \(\frac{527}{110}\)), showing that for x = -15, y is \(\frac{527}{110}\).

Ordered pairs are important as they give us a practical way to visualize a linear equation. Each solution (or pair) corresponds to a point on the graph of the equation, helping us understand the relationship between x and y.

A solution of an equation with two variables, like the one provided, is a set of values for the variables that makes the equation true. In our example, the equation is a balance, represented mathematically as \(35x + 110y = 2\).

Finding the solutions involves different steps, like choosing specific \( x \) values and calculating the corresponding y, which forms ordered pairs. Our process was:

• Substituting given \( x \) values into the equation.
• Solving for y to complete the ordered pair.

For example, substituting \( x = -10 \) gives \(( y = \frac{176}{55} )\), forming the ordered pair \((-10, \frac{176}{55})\). Each substitution and result provides a solution to the equation.

Solutions can be infinite for linear equations with two variables, represented graphically as a straight line. Every point on the line is a solution.

Solving for variables in an equation is a key mathematical skill. This involves isolating one variable on one side of the equation, allowing us to express it as a function of the other variables.

In our example, the process began with the equation: \(35x + 110y = 2\). By rearranging, we solved for y in terms of x. This isolation step used basic algebra:

• Subtract \(35x\) from both sides: \(110y = 2 - 35x\).
• Divide everything by 110 to solve for y: \(y = \frac{2 - 35x}{110}\).

Once y is expressed in terms of x, it becomes straightforward to substitute different x values. This technique is essential not only for creating ordered pairs but also for comprehending how changes in one variable affect others, providing valuable insight into the properties of the line represented by the equation.