Exercises: Intersection of two lines

Bestimme den Schnittpunkt beider Geraden und zeichne diesen in ein Koordinatensystem.

$f (x) = - 3 x + \frac{5}{4}; g (x) = - x - 1$

For this task you need the following basic knowledge: Linear function
Set the functions $f (x)$ and $g (x)$ equal to obtain the intersection point.
$- 3 x + \frac{5}{4}$ $=$ $- x - 1$ $+ 3 x + 1$
$- x + 3 x$ $=$ $\frac{5}{4} + 1$
↓
add
$2 x$ $=$ $\frac{9}{4}$ $: 2$
$x$ $=$ $\frac{9}{8}$
Insert $x$ into one of the two lines equations, for example $g$ :
$y$ $=$ $- \frac{9}{8} - 1$
↓
subtract
$y$ $=$ $- 2.125$
$\Rightarrow S (1.25 / - 2.125)$
Sketch
$f : 2 y - x = 3; g (x) = - \frac{1}{2} x + 4$

For this task you need the following basic knowledge: Linear function
Set the functions $f (x)$ and $g (x)$ equal to obtain the intersection point.
First solve the function $f$ for $y$ .
$2 y - x$ $=$ $3$ $+ x$
$2 y$ $=$ $3 + x$ $: 2$
$y$ $=$ $\frac{3}{2} + \frac{x}{2}$
Now set f and g equal
$\frac{3}{2} + \frac{x}{2}$ $=$ $- \frac{1}{2} x + 4$ $+ \frac{x}{2} - 1.5$
$\frac{x}{2} + \frac{x}{2}$ $=$ $4 - 1.5$
↓
add and subtract
$x$ $=$ $2.5$
Plug $x$ into one of the two line equations, for example $f$ :
$y$ $=$ $\frac{3}{2} + \frac{2.5}{2}$
↓
add
$y$ $=$ $2.75$
$\Rightarrow S (2.5 / 2.75)$
$f (x) = - \frac{2}{3} x - 1; g (x) = \frac{1}{6} x - 4$

For this task you need the following basic knowledge: Linear function
Set the functions $f (x)$ and $g (x)$ equal to obtain the intersection point.
$- \frac{2}{3} x - 1$ $=$ $\frac{1}{6} x - 4$ $+ \frac{2}{3} x + 4$
$\frac{1}{6} x + \frac{2}{3} x$ $=$ $- 1 + 4$
↓
Addiere.
$\frac{5}{6} x$ $=$ $3$ $: \frac{5}{6}$
$x$ $=$ $3 : \frac{5}{6}$
↓
divide by a fraction $\to$ multiply by the inverse
$x$ $=$ $3 \cdot \frac{6}{5}$
↓
multiply
$x$ $=$ $\frac{18}{5}$
Insert $x$ into one of the two lines equations, for example $g$ :
$y$ $=$ $\frac{1}{6} \cdot \frac{18}{5} - 4$
↓
multiply
$y$ $=$ $\frac{3}{5} - 4$
↓
subtract
$y$ $=$ $- 3.4$
$\Rightarrow S (3.6 / - 3.4)$
$f : x = 2; g (x) = - \frac{3}{4} x - \frac{3}{2}$

For this task you need the following basic knowledge: Linear function
Plug $x = 2$ into $g (x)$
$y$ $=$ $- \frac{3}{4} \cdot 2 - \frac{3}{2}$
↓
multiply
$y$ $=$ $- \frac{3}{2} - \frac{3}{2}$
↓
subtract
$y$ $=$ $- \frac{6}{2}$
↓
divide
$y$ $=$ $- 3$
$\Rightarrow S (2 / - 3)$

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$- 3 x + \frac{5}{4}$	$=$	$- x - 1$	$+ 3 x + 1$
$- x + 3 x$	$=$	$\frac{5}{4} + 1$
	↓	add
$2 x$	$=$	$\frac{9}{4}$	$: 2$
$x$	$=$	$\frac{9}{8}$

$2 y - x$	$=$	$3$	$+ x$
$2 y$	$=$	$3 + x$	$: 2$
$y$	$=$	$\frac{3}{2} + \frac{x}{2}$

$\frac{3}{2} + \frac{x}{2}$	$=$	$- \frac{1}{2} x + 4$	$+ \frac{x}{2} - 1.5$
$\frac{x}{2} + \frac{x}{2}$	$=$	$4 - 1.5$
	↓	add and subtract
$x$	$=$	$2.5$

$- \frac{2}{3} x - 1$	$=$	$\frac{1}{6} x - 4$	$+ \frac{2}{3} x + 4$
$\frac{1}{6} x + \frac{2}{3} x$	$=$	$- 1 + 4$
	↓	Addiere.
$\frac{5}{6} x$	$=$	$3$	$: \frac{5}{6}$
$x$	$=$	$3 : \frac{5}{6}$
	↓	divide by a fraction $\to$ multiply by the inverse
$x$	$=$	$3 \cdot \frac{6}{5}$
	↓	multiply
$x$	$=$	$\frac{18}{5}$

$y$	$=$	$\frac{1}{6} \cdot \frac{18}{5} - 4$
	↓	multiply
$y$	$=$	$\frac{3}{5} - 4$
	↓	subtract
$y$	$=$	$- 3.4$

$y$	$=$	$- \frac{3}{4} \cdot 2 - \frac{3}{2}$
	↓	multiply
$y$	$=$	$- \frac{3}{2} - \frac{3}{2}$
	↓	subtract
$y$	$=$	$- \frac{6}{2}$
	↓	divide
$y$	$=$	$- 3$

Exercises: Intersection of two lines

Sketch