A system of linear equations is just a set of two or more linear equations.
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:
- The lines intersect at zero points. (The lines are parallel.)
- The lines intersect at exactly one point. (Most cases.)
- The lines intersect at infinitely many points. (The two equations represent the same line.)
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:
- The Graphing Method . This is useful when you just need a rough answer, or you're pretty sure the intersection happens at integer coordinates. Just graph the two lines, and see where they intersect!
- The Substitution Method . First, solve one linear equation for y in terms of x . Then substitute that expression for y in the other linear equation. You'll get an equation in x . Solve this, and you have the x -coordinate of the intersection. Then plug in x to either equation to find the corresponding y -coordinate. (If it's easier, you can start by solving an equation for x in terms of y , also – same difference!)
- The Linear Combination Method , aka The Addition Method , aka The Elimination Method. Add (or subtract) a multiple of one equation to (or from) the other equation, in such a way that either the x -terms or the y -terms cancel out. Then solve for x (or y , whichever's left) and substitute back to get the other coordinate.
- The Matrix Method . This is really just the Linear Combination method, made simpler by shorthand notation.
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.