How to find a value that creates infinite solutions in a system of equations in 3 variables

[ausmathgenius420]

Hi there,

I'm stuck on this question from my textbook I'm hoping someone can help me.

For which value of a will the following system of equations, corresponding to three planes, have solutions?
[math]2x + y + 3z = 1 \\ x − 3y − z = 5\\ 3x − 2y + 2z = a[/math]
I have tried using reduced row echelon form (I got [imath]\frac{20}{11}z=\frac{7}{11}(a-15)[/imath] ) however trying to cancel the z term would be difficult with the fraction and likely not lead to the correct answer.

I've also tried cancelling z using eqn1 and eqn2 and then cancelling z for eqn 2 and 3 and using those 2 equations... that also doesn't work.

Please helppp!

PS the answer is 6

 

[lev888]

Hi there,

I'm stuck on this question from my textbook I'm hoping someone can help me.

For which value of a will the following system of equations, corresponding to three planes, have solutions?
[math]2x + y + 3z = 1 \\ x − 3y − z = 5\\ 3x − 2y + 2z = a[/math]
I have tried using reduced row echelon form (I got [imath]\frac{20}{11}z=\frac{7}{11}(a-15)[/imath] ) however trying to cancel the z term would be difficult with the fraction and likely not lead to the correct answer.

I've also tried cancelling z using eqn1 and eqn2 and then cancelling z for eqn 2 and 3 and using those 2 equations... that also doesn't work.

Please helppp!

PS the answer is 6

Eliminate one variable. Now you have 2 equations, 2 variables. That's 2 lines. The question becomes: for which values of a the 2 equations represent the same line. Tweak the equations so they are in the _same_ form (e.g. slope-intercept). Then just find a for which slopes and intercepts are the same.

 

[Dr.Peterson]

For which value of a will the following system of equations, corresponding to three planes, have solutions?
[math]2x + y + 3z = 1 \\ x − 3y − z = 5\\ 3x − 2y + 2z = a[/math]
I have tried using reduced row echelon form (I got [imath]\frac{20}{11}z=\frac{7}{11}(a-15)[/imath] ) however trying to cancel the z term would be difficult with the fraction and likely not lead to the correct answer.

The method you suggest worked fine for me. (If you don't like fractions, there are ways to avoid them, though perhaps not what you were taught. But they didn't give me much trouble.)

Please show your work with this method, so we can see where you made a mistake.