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CHAPTER 5 KEY FORMULAS AND EQUATIONS

Linear equation in two unknowns	$a x + b y = c$ (5.1)
Definition of slope	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ (5.2)
Slope-intercept form	$y = m x + b$ (5.3)
System of two linear equations	$\begin{array}{l} a_{1} x + b_{1} y = c_{1} \\ a_{2} x + b_{2} y = c_{2} \end{array}$ (5.4)
Second-order determinant	$\| \begin{array}{l} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array} \| = a_{1} b_{2} - a_{2} b_{1}$ (5.8)
Cramer’s rule	$x = \frac{\| \begin{array}{l} c_{1} & b_{1} \\ c_{2} & b_{2} \end{array} \|}{\| \begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix} \|} and y = \frac{\| \begin{matrix} a_{1} & c_{1} \\ a_{2} & c_{2} \end{matrix} \|}{\| \begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix} \|}$ (5.9)
System of three linear equations	$\begin{array}{l} a_{1} x + b_{1} y + c_{1} z = d_{1} \\ a_{2} x + b_{2} y + c_{2} z = d_{2} \\ a_{3} x + b_{3} y + c_{3} z = d_{3} \end{array}$ (5.10)
Third-order determinant	$\| \begin{array}{l} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} \| = a_{1} b_{2} c_{3} + a_{3} b_{1} c_{2} + a_{2} b_{3} c_{1} - a_{3} b_{2} c_{1} - a_{1} b_{3} c_{2} - a_{2} b_{1} c_{3}$ (5.12)
Cramer’s rule	$x = \frac{\| \begin{array}{l} d_{1} & b_{1} & c_{1} \\ d_{2} & b_{2} & c_{2} \\ d_{3} & b_{3} & c_{3} \end{array} \|}{\| \begin{array}{l} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} \|} y = \frac{\| \begin{array}{l} a_{1} & d_{1} & c_{1} \\ a_{2} & d_{2} & c_{2} \\ a_{3} & d_{3} & c_{3} \end{array} \|}{\| \begin{array}{l} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} \|} z = \| \begin{array}{l} a_{1} & b_{1} & d_{1} \\ a_{2} & b_{2} & d_{2} \\ a_{3} & b \end{array}$

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