A system of linear equations is just a set of two or more linear equations. Show
In two variables ( x and y ) , the graph of a system of two equations is a pair of lines in the plane. There are three possibilities:
Zero solutions: y = − 2 x + 4 y = − 2 x − 3  One solution: y = 0.5 x + 2 y = − 2 x − 3 Infinitely many solutions: y = − 2 x − 4 y + 4 = − 2 x
There are a few different methods of solving systems of linear equations:
See the second graph above. The solution is where the two lines intersect, the point ( − 2 , 1 ) . Example 1: Solve the system { 3 x + 2 y = 16 7 x + y = 19 Solve the second equation for y . y = 19 − 7 x Substitute 19 − 7 x for y in the first equation and solve for x . 3 x + 2 ( 19 − 7 x ) = 16 3 x + 38 − 14 x = 16 − 11 x = − 22 x = 2 Substitute 2 for x in y = 19 − 7 x and solve for y . y = 19 − 7 ( 2 ) y = 5 The solution is ( 2 , 5 ) .Example 2: Solve the system { 4 x + 3 y = − 2 8 x − 2 y = 12 Multiply the first equation by − 2 and add the result to the second equation. − 8 x − 6 y = 4 8 x − 2 y = 12 _ − 8 y = 16 Solve for y . y = − 2 Substitute for y in either of the original equations and solve for x . 4 x + 3 ( − 2 ) = − 2 4 x − 6 = − 2 4 x = 4 x = 1 The solution is ( 1 , − 2 ) .
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What are systems of equations?A system of equations is a set of one or more equations involving a number of variables.The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection. Systems of linear equations are a common and applicable subset of systems of equations. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. The system is said to be inconsistent otherwise, having no solutions. Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent). More general systems involving nonlinear functions are possible as well. These possess more complicated solution sets involving one, zero, infinite or any number of solutions, but work similarly to linear systems in that their solutions are the points satisfying all equations involved. Going further, more general systems of constraints are possible, such as ones that involve inequalities or have requirements that certain variables be integers. Solving systems of equations is a very general and important idea, and one that is fundamental in many areas of mathematics, engineering and science. What are the 3 solutions of systems of linear equation?There are three ways to solve systems of linear equations in two variables: graphing. substitution method. elimination method.
What is the solution of the system of equations explain?The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system. The solution is the ordered pair(s) common to all lines in the system when the lines are graphed. Lines that cross at a point (or points) are defined as a consistent system of equations.
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The solution of a system of linear equations is
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