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Here 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.
A system may contain any number of equations. To determine unique values for , however, the equations must provide three independent constraints. A common square form is:
Example:
In the substitution method, express one variable in terms of another, then insert that expression into the other equation. The following example shows each step.
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 : Substitute the calculated value to find the other variable.
Substitute the value from equation () into equation ():
The solution is and .
In the elimination method, add or subtract two equations to remove one variable. The following calculation solves 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 ():
The solution is and .
Verify:
Solve the system:
The second equation gives . Substitute this expression into the first equation:
Back-substitute into :
Both original equations confirm the result:
The solution is .
To turn the quantities and relationships in a word problem into a linear equation system:
In basketball, there are three types of shots with different point values: free throws ( point), two-point shots ( points), and three-point shots ( points).
Define:
If Wijaya scored points, made shots total with of them being free throws, then:
Substituting into the second equation:
Substitute into the first equation:
From these two equations:
Using elimination or substitution, we get and .
Wijaya made free throws, -point shots, and -point shot.
Linear equation systems have three possible solution types:
In three dimensions, one linear equation represents a plane. Two distinct nonparallel planes intersect in a line. For three or more planes, the common solution set may be a point, a line, a plane, or empty.
Published: . Updated: .
