A system of linear equations is a set of equations with the same variables. To solve a linear system is to find values for the variables that make all of the equations true.

There are a number of ways to solve a system of linear equations. If there are only two variables, you can graph the set of equations on the same coordinate axes and find the intersection if there is one. You can also use various matrix techniques including Gaussian elimination, Cramer's rule, and inverse matrices. You could also try a guess and revise strategy.

Two symbolic techniques often taught are substitution and linear combinations.

(1) Using substitution, we solve one of the equations for one of the variables. Then we substitute the expression into the remaining equation(s) until we have a single equation with one unknown which we can solve by using basic inverse operations.

For example:

9x+12y=-18

10x+12y=-16

We can solve the first equation for x:

9x=-12y-18 ==> x= -4/3y-2

We then substitute this expression for x into the second equation:

10(-4/3y-2)+12y=-16

and solve using basic inverse operations:

-40/3y-20+12y=-16

-4/3y=4

y=-3

Once we know y, substitute into either original equation to get x:

9x+12(-3)=-18 ==> 9x=18 ==> x=2

So the solution is the point (2,-3)

(2) Linear combinations -- sometimes called the multiplication and addition method or elimination method:

Here we add multiples of the equations in such a way that a variable is eliminated. Using the previous example:

9x+12y=-18

10x+12y=-16

We can multiply the first equation by -1 and then add the result to the second equation. (This is equivalent to subtracting the first equation from the second.)

x+0y=2 so x=2 and substituting x=2 into one of the original equations yields y=-3 as before.

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Two symbolic methods are substitution and linear combinations

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