algebra and finite substitution

Question

Use Gauss-Jordan row reduction to solve the given system of equations. HINT [See Examples 1-6.] (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of x, where
y = y(x).)
9x − 2y = 9
54x − 12y = 54

Answer 1

In matrix form it would look like

[9    -2     9]

[54  -12  54]

 

Gaussian elimination involves multiplying a row by a constant and adding to another row so that one of the variables cancels. In this case like most cases we want to eliminate the first variable the x. We can try to get rid of either 9 or 54, where 54 is a better choice because we won't have to multiply by a fraction (i.e. 54/9=6 or 9/54=1/6) where it's easier to multiply the first row by 6 than it is to multiply the second row by 1/6. Also note that we could just subtract one row from another or we could multiply by a negative (i.e. -6)

 

[9 -2 9](-6)=  [-54 12 -54]

Which you'll notice is exactly the same as the second equation, just negative. so if we add them

 

[54 -12 54]+

[-54 12 -54]=

[0    0     0 ]

 

Since they are equal (after multiplying by a constant) and add to zero, that tells you they are actually the same equation and both provides no more information that either one. Since you only have one equation but two variables, this is an unsolvable problem.