# Math Help - consistent linear equations

1. ## consistent linear equations

If you have matrix 2,-3,h
-6 9 5, and you want a value for h so that this matrix is of a consistent linear equation, then what is the value for h?

My answer is that the system of equations is inconsistent no matter what the value of h is.

2. Hello, lord12

If you have matrix: . $\begin{bmatrix}2 &\text{-}3 &h \\ \text{-}6 & 9 & 5\end{bmatrix}$ and you want a value for $h$

so that this matrix is of a consistent linear equation, then what is the value for $h$ ?

We have: . $\left[\begin{array}{cc|c} 2 & \text{-}3 & h \\ \text{-}6 & 9 & 5 \end{array}\right]$

$\begin{array}{c}\\ R_3 + 3R_1\end{array}\left[\begin{array}{cc|c} 2 & \text{-}3 & h \\ 0 & 0 & 3h+5 \end{array}\right]$

If $3h+5 \:\neq\:0$, the system is inconsistent.

If $3h+5\:=\:0 \quad\Rightarrow\quad h \:=\:-\tfrac{5}{3}$, the system is consistent.
. . And there is an infinite number of solutions.

The solutions are: . $\begin{array}{c}x \:=\:\frac{3}{2}t + \frac{h}{2} \\ y \:=\:t \end{array} \quad\hdots$ for any value of $t.$