91Ó°ÊÓ

Linear equations are equations where each term is either a constant or the product of a constant and a single variable.
These equations form straight lines when graphed.
For example, in the given exercise, we have two linear equations:
\[ \frac{1}{2} = 3x - y \] and \[ 2x + 3y = \frac{5}{4} \].
Both equations represent straight lines on a graph.
Solving these equations usually involves finding the point where the lines intersect.

A system of equations consists of multiple equations that share the same set of variables.
To solve these systems, we need to find values for the variables that satisfy all equations simultaneously.
In the provided exercise, the system comprises:

• \[ \frac{1}{2} = 3x - y \]
• \[ 2x + 3y = \frac{5}{4} \]

The solution to this system is a pair of values \((x, y)\) that make both equations true together.
Different methods can be used, like substitution or elimination, to find the solution.

There are two main methods to solve systems of equations: substitution and elimination.
The substitution method involves solving one of the equations for one variable and then substituting this value into the other equation.
In the provided exercise, we used substitution by first solving the first equation for \(y\):
\[ y = 3x - \frac{1}{2} \]
Then, we substituted \(y\) into the second equation and solved for \(x\).
The elimination method involves adding or subtracting equations to eliminate a variable, making it easier to solve for the remaining variable.
Both methods can lead to the solution, but substitution is particularly effective when one equation is easily solved for one variable.
Understanding these methods equips you with the tools to handle most systems of linear equations successfully.