Systems of Linear Equations

Solve the following system of linear equations using the substitution method: \[\begin{align*} x + 2y &= 7 \\ 3x - y &= 5 \end{align*}\]

2. Resolva utilizando a Regra de Crammer (Determinante \[ \left\{\begin{array}{c} 3 x-2 y+z=6 \\ x+y-z=4 \\ 2 x+y-2 z=6 \end{array}\right. \]

Solve the following system of linear equations: \[3x - 2y + z = 1\] \[2x + y - 3z = -1\] \[x + y + z = 3\]

Discussion Assignment Your discussion response for this unit will consist of two parts. First, create 3 equations of the form $a x+b y+c z=d$, where $a, b, c$, and $d$ are constants (integers between -5 and 5). For example, $x+2 y-z=-1$. Perform row operations on your system to obtain a row-echelon form and the solution. Go to the 3D calculator website GeoGebra: www.geogebra.org/3d?lang Ept and enter each of the equations. After you have completed this first task, choose one of the following to complete your discussion post. 1. Reflect on what the graphs are suggesting for one equation, two equations, and three equations, and describe your observations. Think about the equation as a function $f$ of $x$ and $y$, for example, $x+2 y+1=z$ in the example above. Geogebra automatically interprets this way, that is, like $z=f(x, y)=x+2 y+1$, it isolates $z$ in the equation. 2. What did the graphs show when you entered the second equation?

\( \begin{array}{c}3 x-y+z+7 w=13 \\ -2 x+y-z-3 w=-9 \\ -2 x+y-7 w=-8\end{array} \)

\[ \begin{array}{l} x+2 y-3 z=2 \\ x-3 y-z=3 \\ -x+3 y+(4-k) z=7 \end{array} \] If we don't apply the Cramer's method to the given linear system then find the value for $k$.