Hi
How would I be able to find the value of $\displaystyle a$ that would give the equations an infinite number of solutions
$\displaystyle x-4y+z=14$
$\displaystyle -4x-3y+3z=8$
$\displaystyle 17x-30y+az=110$
Thanks
Hi
How would I be able to find the value of $\displaystyle a$ that would give the equations an infinite number of solutions
$\displaystyle x-4y+z=14$
$\displaystyle -4x-3y+3z=8$
$\displaystyle 17x-30y+az=110$
Thanks
1 x (-4) - 2
19 y -7z =-64 5
1 x 17 - 3
-38 y +(17-a)z=128 6
Divide equation 6 by -2 : 19y - (17-a)/2 = -64
By comparing equation 6 to equation 5, we see that (17-a)/2=7 so a=3.
In a case where you are given 3 equations and only two of them are useful, the system is said to have infinite solutions.
Hello, bobred!
How would I be able to find the value of $\displaystyle \,a$
that would give the equations an infinite number of solutions?
. . $\displaystyle \begin{array}{ccc}x-4y+z &=& 14 \\
\text{-}4x-3y+3z&=&8 \\
17x-30y+az &=& 110 \end{array}$
A system of equations has no solution or an infinite number of solutions
. . if the determinant of its coefficients is zero.
$\displaystyle D \;=\; \begin{vmatrix}1 & \text{-}4 & 1 \\ \text{-}4 & \text{-}3 & 3 \\ 17 & \text{-}30 & a \end{vmatrix} \;=\;1\begin{vmatrix}\text{-}3&3\\\text{-}30 &a \end{vmatrix} + 4\begin{vmatrix}\text{-}4 & 3\\17&a\end{vmatrix} + 1\begin{vmatrix}\text{-}4&\text{-}3\\17&\text{-}30\end{vmatrix} $
. . .$\displaystyle =\;(\text{-}3a + 90) + 4(\text{-}4a-51) + (120 + 51) \;=\;\text{-}19a + 57$
If $\displaystyle D = 0$, then: .$\displaystyle \text{-}19a + 57 \:=\:0 \quad\Rightarrow\quad \boxed{a \:=\:3}$
We have: .$\displaystyle \begin{Bmatrix} x - 4y + z &=& 14 & [1] \\
\text{-}4x - 3y + 3z &=& 8 & [2] \\
17x - 30y + 3z &=& 110 & [3] \end{Bmatrix}$
We see that: .$\displaystyle 9\1\cdot\1\text{[equation 1]} - 2\!\cdot\1\text{[equation 2]} \;=\;\text{[equation 3]}$
The system of equations is dependent.
There is an infinite number of solutions.
Let me add to Plato's request that you start showing some attempt of your own on these problems!
Note that after Soroban found "a" that made the determinant 0, he then checked that the equations, with that a, were dependent. If the determinant of coefficients of a system is 0, then either there are an infinite number of solutions or there is no solution.