# Thread: Algebra - Linear systems

1. ## Algebra - Linear systems

1.
a) For what value of a is there no unique solution for the following simultaneous equations.
3x - 2y = 5
5x + ay = 4

b) When a has this value is there an infinite number of solutions or no solution? Justify your answer.

2.
Solve for x: 3 - x/4 < 2 - 3x/5

2. Originally Posted by LilDragonfly
1.
a) For what value of a is there no unique solution for the following simultaneous equations.
3x - 2y = 5
5x + ay = 4

b) When a has this value is there an infinite number of solutions or no solution? Justify your answer.

2.
Solve for x: 3 - x/4 < 2 - 3x/5
1.
That happens when the determinant is zero, thus,
$\displaystyle \left| \begin{array}{cc}3&-2\\5&a \end{array} \right|=0$
Thus,
$\displaystyle 3a+10=0$
Thus,
$\displaystyle a=-\frac{10}{3}$.

To determine if this system is consistent or inconsistent substitute the value of $\displaystyle a$ yields,
$\displaystyle \left\{ \begin{array}{c} 3x-2y=5\\15x-10y=12$

Note:I multiplied to second equation by 3 to clear denominator.

Now, multiply the first equation by 5 to yield,
$\displaystyle \left\{ \begin{array}{c} 15x-10y=25\\15x-10y=12$

How can $\displaystyle 15x-10y=25\mbox{ and }12$?
Impossible, thus this system is inconsistent.

3. 2.
You have,
$\displaystyle 3-x/4<2-3x/5$
Multiply by LCD=20,
$\displaystyle 60-5x<40-12x$
$\displaystyle 60<40-7x$
$\displaystyle 20<-7x$
$\displaystyle -20/7>x$