Solve: 0.5x + y = -3.5, x - 3.2y = 3.4.

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The solution to the system of equations is `x=-37/35, y=-104/35`

We are given a system of linear equations:

.5x + y = -3.5
x - 3/2y = 3.4

There are a number of ways to solve a system of two linear equations in two variables.

1) We can use linear combinations (often called the multiplication/addition method or the elimination method). If we choose to "eliminate" the x, we can multiply the first equation by 2 and the second equation by -1 to get this system:

x + 2y = -7
-x + 1.5y = -3.4

We can add these equations together to get a third line that includes the solution point; then we get 3.5y = -10.4. Dividing both sides by 3.5 we get `y=-2.9bar(714285)` or `y=-104/35`

Substituting into the first equation gives `x-208/35=-7 ==> x=-35/37` so the solution point is `(-35/37,-104/35) `

** Note that such a system either has exactly one solution, no solutions, or an infinite number of solutions.

(2) We can use the substitution method. Solving the first equation for y yields `y=-.5x-3.5` . We can substitute this expression for y into the second equation:

`x-1.5(-.5x-3.5)=3.4 ==> x+.75x+5.25=3.4 ==> 1.75x=-1.85`

Then `x=-1.85/1.75=-1.0bar(571428)=-37/35` and `y=-.5x-3.5 ==> y=-.5(-37/35)-3.5=-104/35`

(3) Other methods include graphing (very inefficient here), inverse matrices, Gaussian elimination, and Cramer's Rule, among others.

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