Infinite Solutions????

[Novice]

I found this question, and I'm stuck....

Let b be a positive number such that the system
ax+3y=1
5x+ay=b
has an infinite number of solutions. By rounding to the nearest hundredth, the value of b equals...
A) 0.60
B) 1.29
C) 1.67
D) 3.87

I cannot tell you exactly where I am stuck as I do not know how to proceed, having never faced this type of question before. Nor do I know when will the equation have infinite solutions. I have solved the other questions of the exercise, so I think I know the theory to solve this one as well. But I can't, so Please Help!

 

[soroban]

Hello, Novice!

I found this question, and I'm stuck....

Let b be a positive number such that the system

. . \(\displaystyle \begin{Bmatrix}ax+3y&=&1 & [1] \\ 5x+ay&=&b & [2] \end{Bmatrix}\) .has an infinite number of solution

Find the value of b to the nearest hundredth.,

. . (A) 0.60 . . (B) 1.29 . . (C) 1.67 . . (D) 3.87

The system has an infinite number of solutions if one equation is a multiple of the other.
. . (They are actually the same equation, but somewhat disguised.)


\(\displaystyle \begin{array}{ccccccc}\text{Multiply [1] by }b\!: & abx + 3by &=& b \\ \text{Compare to [2]:} & 5x + ay &=& b \end{array}\)


These two equations are supposed to be identical.
. . Their corresponding coefficients are equal.. \(\displaystyle \begin{Bmatrix}ab &=& 5 & [3] \\ 3b &=& a & [4] \end{Bmatrix}\)

From [3]: .\(\displaystyle ab \:=\:5 \quad\Rightarrow\quad a \:=\:\dfrac{5}{b}\)

Substitute into [4]: .\(\displaystyle 3b \:=\:\dfrac{5}{b} \quad\Rightarrow\quad b^2 \:=\:\dfrac{5}{3} \quad\Rightarrow\quad b \:=\:\sqrt{\dfrac{5}{3}} \)


Therefore: .\(\displaystyle b \;=\;1.290994449 \;\approx\;1.29 \;\;\text{ Answer (B)}\)

 

[Novice]

Thanks a lot! I can sleep now!

 

[mmm4444bot]

Nor do I know when will [a system of linear equations] have infinite solutions

Then you need to go back to your textbook :!:

Soroban may put you to sleep, but he certainly has not put this issue to rest, for you.

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"Spoon feeding, in the long run, teaches us nothing but the shape of the spoon." ~ Edward Morgan Forster