Unit 5 SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES
There are many uses for the fundamental mathematical concepts of linear equations and inequalities. Systems of linear equations and inequalities arise whenever we must solve multiple equations or inequalities simultaneously. These frameworks provide a solid basis for modeling realistic circumstances and solving challenging problems. Methods, properties, and applications for solving systems of linear equations and inequalities are discussed in this article.
Table of Contents
Solutions to Systems and Solving by Graphing
Systems of linear equations in mathematics are a grouping of two or more equations using the same variables. One of the essential ideas of algebra is finding the solutions to these systems. Graphing is one way to solve systems of linear equations.
Each equation is represented as a straight line on a coordinate plane when systems are solved by graphing. The solution to the system can be found at the point where the lines meet. Only one solution exists if the lines intersect in a single point. If the lines do not cross, then the system cannot be solved. The system has infinitely many solutions when the lines overlap.
Systems of linear equations can be graphed to show the solution visually. For systems with uncomplicated equations and integer solutions, it is an easy procedure. However, other approaches, such as substitution or elimination, may be more effective for more complex systems or when dealing with non-integer solutions.
Examples#1 (unique solution) Solve the system of linear equations by using the graphical method.
2x+3y=1
x + y =0
[answer]
Examples#2 (no solution) Solve the system of linear equations by using the graphical method.
x+2y=6
2x+4y=2
[answer]
Examples#2 (no solution) Solve the system of linear equations by using the graphical method.
x-2y=2
3x-6y=6
[answer]
Solving Systems by Substitution:
Substitution is another linear equation solution method. One equation is solved for a single variable and utilized as a stand-in for the other(s).
Solve a system by substitution by solving one equation for one variable in terms of the other. Insert the variable’s expression into the other equations. Swapping the variable turns the system into a single equation with one variable. Solve this equation and add the answer to the original equation to find the variable’s value.
Substitution makes it easy to substitute a variable-solved problem. It is a versatile approach that works with systems that contain any number of equations. When dealing with complex systems or equations that are challenging to answer explicitly, it might become laborious.
Examples#1 (unique solution) Solve the system of linear equations by using the substitution method.
2x+3y=1
x +y=0
[answer]
Examples#2 (no solution) Solve the system of linear equations by using the substitution method.
x+2y=6
2x+4y=2
[answer]
Examples#2 (no solution) Solve the system of linear equations by using the substitution method.
x-2y=2
3x-6y=6
[answer]
Solving system by Cramer’s rule
Determinants help find the right answer. This method works well for systems of equations with the same number of variables.
Cramer’s Rule says that a system of equations has only one solution if and only if the matrix of coefficients’ determinant (D) is non-zero. Cramer’s Rule solves linear equations with few variables efficiently.
Using determinants, Cramer’s Rule provides a methodical procedure for locating the solution to a system of linear equations. In subjects such as mathematics, engineering, and physics where systems of equations must be analyzed and solved, this method is invaluable.
Examples#1 Solve the system of linear equations by using the Cramer’s Rule.
2x+3y=1
x+y=0
[answer]