3. Solve the following system of equations by



and check your answer



a) Graphing.


Graph both equations. Solve the equations for y. In the first one



transpose the x term



and divide both sides by 3.



Do the same thing for the other equation






and divide by 2.



Now make up tables of x's and y's for both equations.




plot the points and draw the graphs. We see that the point of

intersection is (3, -1), so the solution is






If we substitute these numbers into the original equations, in

the first equation we get


2(3) + 3(-1) = 3


6 - 3 = 3



and in the second equation we get


3(3) + 2(-1) = 7


9 - 2 = 7


and the numbers check in both equations.


b. Substitution



Solve one of the equations for one of the unknowns. In this

case it is easiest to solve the first equation for y.



Now substitute this solution into the other equation.



Remove parentheses





At this point we could combine the x terms. To subtract the

coefficients, 3 - 4/3 = 5/3, but it might be easier to first clear

denominators by multiplying both sides of the equation by 3.






To find y, substitute this solution into the equation where we

solved for y as a function of x




c) Addition


Multiply the equations by suitable numbers so that the

coefficients on one of the unknowns match up. In this case, if we

multiply the first equation by 3 and the second equation by -2, the

coefficients of the x's will match up and they will go away when we




To find x, we substitute this value of y into either of the

original equations. The first equation might be easiest.





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