91Ó°ÊÓ

Linear equations are equations of the first order, which means the variables are raised to the power of one. When solving linear equations, we often encounter a system of linear equations, which involves finding the values of variables that satisfy multiple linear equations simultaneously. In the exercise, we aim to find the price of citrons and fragrant wood apples. We use two variables: let the price of one citron be denoted as \( x \), and the price of one fragrant wood apple be denoted as \( y \). We then set up equations based on the given conditions, leading us to the system of linear equations: \( 9x + 7y = 107 \) and \( 7x + 9y = 101 \). To solve these, we can use methods such as the substitution method or the elimination method. Understanding these techniques will make solving such systems more manageable and clearer.

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. Although not used in this solution, it's essential to understand it for different problems. Here is a general approach to the substitution method:

• Solve one of the equations for one of the variables (e.g., solve for \( x \) in terms of \( y \) or vice versa).
• Substitute that expression into the other equation. This substitution transforms the original system of equations into a single equation with one variable.
• Solve this single-variable equation.
• Substitute the found value back into the expression obtained in the first step to find the value of the other variable.

This process makes it easier to tackle systems with simpler equations and understand interdependencies between variables. For instance, if we had solved \( 9x + 7y = 107 \) for \( x \), we could substitute that into \( 7x + 9y = 101 \) to find \( y \).

The elimination method simplifies the process of solving systems of linear equations by eliminating one variable, allowing you to solve for the other. Here's how it works using our exercise as an example:

• Start with the equations: \( 9x + 7y = 107 \) and \( 7x + 9y = 101 \).
• Adjust the equations so that one of the variables can be eliminated when the equations are subtracted. We do this by multiplying the equations by appropriate numbers. In our case, we multiply the first equation by 7 and the second by 9:
  \[ 63x + 49y = 749 \] and \[ 63x + 81y = 909 \].
• Subtract the first adjusted equation from the second to eliminate \( x \):
  \[ (63x + 81y) - (63x + 49y) = 909 - 749 \]
  which simplifies to \[ 32y = 160 \].
• Solve for \( y \):
  \[ y = 5 \].
• Substitute \( y \) back into one of the original equations to find \( x \):
  \( 9x + 7(5) = 107 \)
  \( 9x + 35 = 107 \)
  \( 9x = 72 \)
  \( x = 8 \).

The elimination method often provides a clearer path to the solution, especially when the equations are amenable to simple multiplication and subtraction.