Exercises: Intersection of two lines

Check whether the following lines $g, h$ and $i$ run through a common point.

$g (x) = x + 1; h : 2 y + x + 4 = 0; 3 y - 5 x = 7$

For this task you need the following basic knowledge: Linear function
First bring all lines equations into the general form $y = m x + t$ .
Line $g$ :
$g (x) = x + 1$
$\Leftrightarrow y = x + 1$
Line $h$ :
$2 y + x + 4 = 0$
$\Leftrightarrow 2 y = - x - 4$
$\Leftrightarrow y = - \frac{1}{2} x - 2$
Line $i$ :
$3 y - 5 x = 7$
$\Leftrightarrow 3 y = 5 x + 7$
$\Leftrightarrow y = \frac{5}{3} x + \frac{7}{3}$
Determine the intersection of $g$ and $h$
Set the respective right sides equal.
$x + 1$ $=$ $- \frac{1}{2} x - 2$ $+ \frac{1}{2} x - 1$
↓
Get all $x$ -terms on one side.
$x + \frac{1}{2} x$ $=$ $- 2 - 1$
↓
Conclude
$\frac{3}{2} x$ $=$ $- 3$ $: \frac{3}{2}$
$x$ $=$ $- 2$
↓
Plug into $g$ to get the $y$ -coordinate.
$y$ $=$ $- 2 + 1 = - 1$
The intersection point is $S_{g h} (- 2 | - 1) .$
Determine the intersection of $g$ and $i$
Set the respective right sides equal.
$x + 1$ $=$ $\frac{5}{3} x + \frac{7}{3}$ $- \frac{5}{3} x - 1$
↓
Get all $x$ -terms on one side.
$x - \frac{5}{3} x$ $=$ $\frac{7}{3} - 1$
↓
Conclude.
$- \frac{2}{3} x$ $=$ $\frac{4}{3}$ $: (- \frac{2}{3})$
$x$ $=$ $- 2$
↓
Plug into $g$ to get the $y$ -coordinate.
$y$ $=$ $- 2 + 1 = - 1$
The intersection point is $S_{g i} (- 2 | - 1) .$
Since $g$ intersects with $h$ and with $i$ at the same point, $h$ and $i$ also intersect at this point.
The lines therefore all run through the common point $(- 2 | - 1)$ .
$g (x) = \frac{1}{6} x + \frac{3}{2}; h (x) = - \frac{2}{3} x + 2; i : 2 x - y = 3$

For this task you need the following basic knowledge: Linear function
First bring all lines equations into the general form $y = m x + t$ .
Line $g$ :
$g (x) = \frac{1}{6} x + \frac{3}{2}$
$\Leftrightarrow y = \frac{1}{6} x + \frac{3}{2}$
Line $h$ :
$h (x) = - \frac{2}{3} x + 2$
$\Leftrightarrow y = - \frac{2}{3} x + 2$
Line $i$ :
$2 x - y = 3$
$\Leftrightarrow y = 2 x - 3$
Determine the intersection of $g$ and $h$
Set the respective right sides equal.
$\frac{1}{6} x + \frac{3}{2}$ $=$ $- \frac{2}{3} x + 2$ $+ \frac{2}{3} x - \frac{3}{2}$
↓
Get all $x$ -terms on one side.
$=$
↓
Conclude.
$\frac{1}{6} x + \frac{2}{3} x$ $=$ $2 - \frac{3}{2}$
$\frac{5}{6} x$ $=$ $\frac{1}{2}$ $: \frac{5}{6}$
$x$ $=$ $\frac{3}{5}$
↓
Plug into $g$ to get the $y$ -coordinate.
$y$ $=$ $\frac{1}{6} \cdot \frac{3}{5} + \frac{3}{2}$
$y$ $=$ $\frac{1}{10} + \frac{3}{2} = \frac{16}{10}$
$y$ $=$ $\frac{8}{5}$
The intersection point is $S_{g h} (\frac{3}{5} | \frac{8}{5}) .$
Determine the intersection of $g$ and $i$
Set the respective right sides equal.
$\frac{1}{6} x + \frac{3}{2}$ $=$ $2 x - 3$ $- 2 x - \frac{3}{2}$
↓
Get all $x$ -terms on one side.
$\frac{1}{6} x - 2 x$ $=$ $- 3 - \frac{3}{2}$
↓
Conclude.
$- \frac{11}{6} x$ $=$ $- \frac{9}{2}$ $: (- \frac{11}{6})$
$x$ $=$ $\frac{9}{2} \cdot \frac{6}{11} = \frac{27}{11}$
↓
Plug into $g$ to get the $y$ -coordinate.
$y$ $=$ $\frac{1}{6} \cdot \frac{27}{11} + \frac{3}{2}$
$y$ $=$ $\frac{9}{22} + \frac{3}{2} = \frac{42}{44}$
$y$ $=$ $\frac{21}{11}$
The intersection point is $S_{g i} (\frac{27}{11} | \frac{21}{11}) .$
Thus the line $g$ intersects the line $h$ at a different point than the line $i$ . So the lines do not run through a common point.

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$x + 1$	$=$	$- \frac{1}{2} x - 2$	$+ \frac{1}{2} x - 1$
	↓	Get all $x$ -terms on one side.
$x + \frac{1}{2} x$	$=$	$- 2 - 1$
	↓	Conclude
$\frac{3}{2} x$	$=$	$- 3$	$: \frac{3}{2}$
$x$	$=$	$- 2$
	↓	Plug into $g$ to get the $y$ -coordinate.
$y$	$=$	$- 2 + 1 = - 1$

$x + 1$	$=$	$\frac{5}{3} x + \frac{7}{3}$	$- \frac{5}{3} x - 1$
	↓	Get all $x$ -terms on one side.
$x - \frac{5}{3} x$	$=$	$\frac{7}{3} - 1$
	↓	Conclude.
$- \frac{2}{3} x$	$=$	$\frac{4}{3}$	$: (- \frac{2}{3})$
$x$	$=$	$- 2$
	↓	Plug into $g$ to get the $y$ -coordinate.
$y$	$=$	$- 2 + 1 = - 1$

$\frac{1}{6} x + \frac{3}{2}$	$=$	$- \frac{2}{3} x + 2$	$+ \frac{2}{3} x - \frac{3}{2}$
	↓	Get all $x$ -terms on one side.
	$=$
	↓	Conclude.
$\frac{1}{6} x + \frac{2}{3} x$	$=$	$2 - \frac{3}{2}$
$\frac{5}{6} x$	$=$	$\frac{1}{2}$	$: \frac{5}{6}$
$x$	$=$	$\frac{3}{5}$
	↓	Plug into $g$ to get the $y$ -coordinate.
$y$	$=$	$\frac{1}{6} \cdot \frac{3}{5} + \frac{3}{2}$
$y$	$=$	$\frac{1}{10} + \frac{3}{2} = \frac{16}{10}$
$y$	$=$	$\frac{8}{5}$

$\frac{1}{6} x + \frac{3}{2}$	$=$	$2 x - 3$	$- 2 x - \frac{3}{2}$
	↓	Get all $x$ -terms on one side.
$\frac{1}{6} x - 2 x$	$=$	$- 3 - \frac{3}{2}$
	↓	Conclude.
$- \frac{11}{6} x$	$=$	$- \frac{9}{2}$	$: (- \frac{11}{6})$
$x$	$=$	$\frac{9}{2} \cdot \frac{6}{11} = \frac{27}{11}$
	↓	Plug into $g$ to get the $y$ -coordinate.
$y$	$=$	$\frac{1}{6} \cdot \frac{27}{11} + \frac{3}{2}$
$y$	$=$	$\frac{9}{22} + \frac{3}{2} = \frac{42}{44}$
$y$	$=$	$\frac{21}{11}$

Exercises: Intersection of two lines

Determine the intersection of g and h

Determine the intersection of g and i

Determine the intersection of g and h

Determine the intersection of g and i

Determine the intersection of $g$ and $h$

Determine the intersection of $g$ and $i$

Determine the intersection of $g$ and $h$

Determine the intersection of $g$ and $i$