Exercises: Linear functions as lines in coordinate systems

$y = 3 x - 2$

Set $y = 0$ to determine the zero. This is, since the line intersects the $x$ -axis exactly where $y = 0$ .

$3 x - 2$	$=$	$0$	$+ 2$
$3 x$	$=$	$2$	$: 3$
$x_{N1}$	$=$	$\frac{2}{3}$
	↓	The first line has a zero at $x_{N} = \frac{2}{3}$ .

Proceed in the same way for the second line.

$y$	$=$	$- \frac{3}{4} x + 1$
	↓	Set y=0 to determine the zero
$- \frac{3}{4} x + 1$	$=$	$0$	$- 1$
$- \frac{3}{4} x$	$=$	$- 1$	$: (- \frac{3}{4})$
$x$	$=$	$\frac{1}{\frac{3}{4}}$
	↓	You divide by a fraction $\to$ multiply by the reciprocal value
$x_{N2}$	$=$	$\frac{4}{3}$
	↓	The first line has a zero at $x_{N} = \frac{4}{3}$ .

Set the two function equations equal. The lines intersect where both have the same $x$ - and the same $y$ -value.

$3 x - 2$	$=$	$- \frac{3}{4} x + 1$	$+ \frac{3}{4} x + 2$
$3 x + \frac{3}{4} x$	$=$	$1 + 2$
$3.75 x$	$=$	$3$	$: 3.75$
$x$	$=$	$\frac{3}{3.75}$
$x_{s}$	$=$	$0.8$
	↓	Plug $x_{s}$ into one of both functions
$y$	$=$	$3 \cdot 0.8 - 2$
$y$	$=$	$2.4 - 2$
$y_{s}$	$=$	$0.4$

$\Rightarrow S (0.8 | 0,4)$

The intersection is at $S (0.8 | 0.4)$ .