91Ó°ÊÓ

Linear equations are fundamental in algebra and appear often in word problems. They typically have one or two variables and form straight lines when graphed. In the exercise, we encounter two linear equations: \(x + y = 40\) and \(y = 4x\). These equations help us describe the relationship between the pounds of oranges \(x\) and apples \(y\). The first equation represents the total weight, whereas the second expresses that the pounds of apples are four times that of pounds of oranges.
This step-by-step process shows how establishing equations from word problems can lead us to solutions.

The substitution method is a powerful technique for solving systems of linear equations. It involves solving one equation for one variable and then substituting this expression into the other equation.
In our problem, we start by solving \(y = 4x\). We then substitute \(4x\) for \(y\) in the first equation:
\[ x + 4x = 40 \ 5x = 40 \ x = 8 \]
This solution of \(x\) is then substituted back into the second equation to find \(y\):
\[ y = 4 \times 8 = 32 \]
This shows how substitution simplifies solving systems of equations by reducing the problem to a single variable.

Variables are symbols, usually letters, that represent unknown numbers in equations. Here, \(x\) stands for the pounds of oranges, and \(y\) for the pounds of apples.
Identifying variables is the first step in solving word problems. We defined our variables and then formulated equations based on those definitions. Using variables allows us to model real-world problems mathematically and find solutions systematically.
Understanding what each variable represents is crucial. It helps us set up correct equations and interpret the solutions properly. In our problem, replacing undefined terms ('apples' and 'oranges') with variables \(x\) and \(y\) made it easier to use algebraic methods to find the amounts.