For AI agents: use /llms.txt for the Nakafa content index.
Imagine you're making a cake recipe. You know the cake requires eggs and flour in specific amounts. However, you only know the total ingredients and their ratios. This is similar to a linear equation system - we're looking for unknown values based on related information.
A linear equation system is a collection of two or more linear equations that must be satisfied simultaneously. Each linear equation has the form:
Where are coefficients, are variables, and is a constant.
A two-variable linear equation system consists of two equations with two variables (usually and ). The general form is:
Example:
The solution to this system is the pair of values that satisfies both equations.
For three variables, we need at least three equations:
Example:
The substitution method works by replacing one variable with another. Let's solve the following example:
Step : Express one variable from the simpler equation.
From equation (): , we express in terms of :
Step : Substitute into the other equation.
Insert equation () into equation ():
Step : Solve the resulting equation.
Step : Back-substitute to find the other variable.
Substitute the value from equation () into equation ():
Therefore, the solution is and .
The elimination method works by eliminating one variable. Let's solve the same example:
Step : Match the coefficients of one variable.
Multiply equation () by to match the coefficient of :
Step : Eliminate the variable by subtracting the equations.
Step : Use the value of to find .
Substitute the value from equation () into equation ():
Therefore, the solution is and .
Verify:
Mathematical modeling is the process of converting real-world problems into mathematical form. For linear equation systems, we:
In basketball, there are three types of shots with different point values: free throws ( point), two-point shots ( points), and three-point shots ( points).
Let's define:
If Wijaya scored points, made shots total with of them being free throws, then:
Substituting into the second equation:
Substituting into the first equation:
From these two equations:
Using elimination or substitution, we get and .
Therefore, Wijaya made free throws, two-point shots, and three-point shot.
Linear equation systems have three possible solution types:
In three dimensions (three variables), a linear equation is represented as a plane. The intersection of two planes forms a line, and the intersection of three planes can form a point.
