Finding Multiple Solutions to Linear Equations in Two Variables
In this section, we delve into the nature of linear equations in two variables, determining how many solutions they possess. Unlike linear equations in one variable which offer a unique solution, linear equations involving two variables present the possibility of infinite solutions due to the presence of two variables, x and y. For instance, considering the equation 2x + 3y = 12, we see that specific pairs of (x, y) values, such as (3, 2), (0, 4), and (6, 0), satisfy this equation. Furthermore, one can generate more solutions by choosing specific values for one variable and subsequently solving for the other. For example, assigning x different values leads to corresponding y values through substitutions. Hence, the section concludes that a linear equation in two variables does not just have limited solutions but an infinite range based on varying values for either variable. This foundational understanding marks the different handling of solving linear equations compared to their one-variable counterparts.