Exercises: Linear functions, zeros, axis intercepts

Transform the equation into the form $y = a x + b$ .

$2 x - y = 6$

$2 x - y$ $=$ $6$ $- 2 x$
$- y$ $=$ $6 - 2 x$ $\cdot (- 1)$
$y$ $=$ $2 x - 6$
$x = \frac{1}{2} (y + 1)$

$x$ $=$ $\frac{1}{2} (y + 1)$ $\cdot 2$
↓
The fraction on the right side vanishes because $\frac{1}{2} \cdot 2 = 1$ .
$2 x$ $=$ $y + 1$ $- 1$
↓
Swap the left and right sides.
$y$ $=$ $2 x - 1$
$\frac{2}{5} y = 2 x - 1$

$\frac{2}{5} y$ $=$ $2 x - 1$ $\cdot \frac{5}{2}$
↓
Note: On the right-hand side, each element is multiplied by $\frac{5}{2}$ :
$\begin{array}{l} \frac{2}{5} \cdot y \cdot \frac{5}{2} = \frac{10}{10} \cdot y = 1 \cdot y; \\ 2 x \cdot \frac{5}{2} = \frac{10}{2} x = 5 x; \\ - 1 \cdot \frac{5}{2} = - \frac{5}{2} \end{array}$
$y$ $=$ $5 x - \frac{5}{2}$
$y = 3 (2 x - 1)$

$y$ $=$ $3 (2 x - 1)$
↓
Multiply out the bracket.
$y$ $=$ $6 x - 3$

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$x$	$=$	$\frac{1}{2} (y + 1)$	$\cdot 2$
	↓	The fraction on the right side vanishes because $\frac{1}{2} \cdot 2 = 1$ .
$2 x$	$=$	$y + 1$	$- 1$
	↓	Swap the left and right sides.
$y$	$=$	$2 x - 1$