Exercises: Intersection of two lines

Calculate the intersection points of a straight line.

Consider are the function equations of the two lines $g_{1} (x)$ and $g_{2} (x)$ . Calculate the intersection of the two lines and draw the lines in a coordinate system.

$g_{1} (x) = \frac{1}{2} x + 2 g_{2} (x) = - \frac{1}{2} x + 4$

For this task you need the following basic knowledge: Linear function
Calculating the line intersection
Set $g_{1} (x)$ and $g_{2} (x)$ equal.
$\frac{1}{2} x + 2$ $=$ $- \frac{1}{2} x + 4$ $| + \frac{1}{2} x - 2$
$\frac{1}{2} x + \frac{1}{2} x$ $=$ $4 - 2$
$x_{S}$ $=$ $2$
↓
Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$ $=$ $1 + 2$
$y_{S}$ $=$ $3$
$\Rightarrow S (x_{S} | y_{S}) = S (2 | 3)$
Sketch
Connect the $y$ -axis intercepts (here $A$ and $B$ ) with the calculated intersection point $S$ . This way you obtain the two lines.
$g_{1} (x) = 2 x - 1 g_{2} (x) = - 2 x + 1$

For this task you need the following basic knowledge: Linear function
Calculating the line intersection
Set $g_{1} (x)$ and $g_{2} (x)$ equal.
$2 x - 1$ $=$ $- 2 x + 1$ $+ 2 x + 1$
$2 x + 2 x$ $=$ $1 + 1$
$4 x$ $=$ $2$ $: 4$
$x_{S}$ $=$ $\frac{1}{2}$
↓
Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$ $=$ $2 \cdot \frac{1}{2} - 1$
$y$ $=$ $1 - 1$
$y_{S}$ $=$ $0$
$\Rightarrow S (x_{S} | y_{S}) = S (\frac{1}{2} | 0)$
Sketch
Connect the $y$ -axis intercepts (here $A$ and $B$ ) with the calculated intersection point $S$ . This way you obtain the two lines.
$g_{1} (x) = \frac{3}{4} x - 4 g_{2} (x) = - \frac{1}{2} x - 1$

For this task you need the following basic knowledge: Linear function
Calculating the line intersection
Set $g_{1} (x)$ and $g_{2} (x)$ equal.
$g_{1} (x)$ $=$ $g_{2} (x)$
$\frac{3}{4} x - 4$ $=$ $- \frac{1}{2} x - 1$ $+ \frac{1}{2} x + 4$
$\frac{3}{4} x + \frac{1}{2} x$ $=$ $- 1 + 4$
$1.25 x$ $=$ $3$ $1.25$
$x$ $=$ $\frac{3}{1.25}$
$x_{S}$ $=$ $2.4$
↓
Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$ $=$ $\frac{3}{4} \cdot 2.4 - 4$
$y$ $=$ $1.8 - 4$
$y_{S}$ $=$ $- 2.2$
$\Rightarrow S (x_{S} | y_{S}) = S (2.4 | - 2.2)$
Sketch
Connect the $y$ -axis intercepts (here $A$ and $B$ ) with the calculated intersection point $S$ . This way you obtain the two lines.
$g_{1} (x) = - \frac{1}{2} x + 2 g_{2} (x) = \frac{1}{2} x + 3$

For this task you need the following basic knowledge: Linear function
Calculating the line intersection
Set $g_{1} (x)$ and $g_{2} (x)$ equal.
$g_{1} (x)$ $=$ $g_{2} (x)$
$- \frac{1}{2} x + 2$ $=$ $\frac{1}{2} x + 3$ $| + \frac{1}{2} x - 3$
$2 - 3$ $=$ $\frac{1}{2} x + \frac{1}{2} x$
$x_{S}$ $=$ $- 1$
↓
Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$ $=$ $- \frac{1}{2} \cdot (- 1) + 2$
$y$ $=$ $\frac{1}{2} + 2$
$y_{S}$ $=$ $2.5$
$\Rightarrow S (x_{S} | y_{S}) = S (- 1 | 2.5)$
Sketch
Connect the $y$ -axis intercepts (here $A$ and $B$ ) with the calculated intersection point $S$ . This way you obtain the two lines.

$g_{1} (x) = \frac{2}{3} x + 2 g_{2} (x) = \frac{1}{2} x + 3$

$g_{1} (x) = \frac{3}{4} x + 1 g_{2} (x) = \frac{1}{2} x + 2$

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$\frac{1}{2} x + 2$	$=$	$- \frac{1}{2} x + 4$	$\| + \frac{1}{2} x - 2$
$\frac{1}{2} x + \frac{1}{2} x$	$=$	$4 - 2$
$x_{S}$	$=$	$2$
	↓	Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$	$=$	$1 + 2$
$y_{S}$	$=$	$3$

$2 x - 1$	$=$	$- 2 x + 1$	$+ 2 x + 1$
$2 x + 2 x$	$=$	$1 + 1$
$4 x$	$=$	$2$	$: 4$
$x_{S}$	$=$	$\frac{1}{2}$
	↓	Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$	$=$	$2 \cdot \frac{1}{2} - 1$
$y$	$=$	$1 - 1$
$y_{S}$	$=$	$0$

$g_{1} (x)$	$=$	$g_{2} (x)$
$\frac{3}{4} x - 4$	$=$	$- \frac{1}{2} x - 1$	$+ \frac{1}{2} x + 4$
$\frac{3}{4} x + \frac{1}{2} x$	$=$	$- 1 + 4$
$1.25 x$	$=$	$3$	$1.25$
$x$	$=$	$\frac{3}{1.25}$
$x_{S}$	$=$	$2.4$
	↓	Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$	$=$	$\frac{3}{4} \cdot 2.4 - 4$
$y$	$=$	$1.8 - 4$
$y_{S}$	$=$	$- 2.2$

$g_{1} (x)$	$=$	$g_{2} (x)$
$- \frac{1}{2} x + 2$	$=$	$\frac{1}{2} x + 3$	$\| + \frac{1}{2} x - 3$
$2 - 3$	$=$	$\frac{1}{2} x + \frac{1}{2} x$
$x_{S}$	$=$	$- 1$
	↓	Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$	$=$	$- \frac{1}{2} \cdot (- 1) + 2$
$y$	$=$	$\frac{1}{2} + 2$
$y_{S}$	$=$	$2.5$

$g_{1} (x)$	$=$	$g_{2} (x)$
$\frac{2}{3} x + 2$	$=$	$\frac{1}{2} x + 3$	$- \frac{1}{2} x - 2$
	↓	Get the fractions on a common denominator.
$\frac{2}{3} x - \frac{1}{2} x$	$=$	$3 - 2$
	↓	subtract
$\frac{4}{6} x - \frac{3}{6} x$	$=$	$3 - 2$
$\frac{1}{6} x$	$=$	$1$	$: \frac{1}{6}$
	↓	dividing by a fraction $\to$ multiplying with the inverse
$x$	$=$	$\frac{1}{\frac{1}{6}}$
$x_{S}$	$=$	$6$
	↓	Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$	$=$	$\frac{2}{3} \cdot 6 + 2$
	↓	add
$y$	$=$	$4 + 2$
$y_{S}$	$=$	$6$

$g_{1} (x)$	$=$	$g_{2} (x)$
$\frac{3}{4} x + 1$	$=$	$\frac{1}{2} x + 2$	$- \frac{1}{2} x - 1$
$\frac{3}{4} x - \frac{1}{2} x$	$=$	$2 - 1$
	↓	Get the fractions on a common denominator.
$\frac{3}{4} x - \frac{2}{4} x$	$=$	$2 - 1$
	↓	subtract
$\frac{1}{4} x$	$=$	$1$	$: \frac{1}{4}$
	↓	dividing by a fraction $\to$ multiplying with the inverse
$x$	$=$	$\frac{1}{\frac{1}{4}}$
$x_{S}$	$=$	$4$
	↓	Plug $x = x_{s}$ into $g_{1} (x)$ .
$y$	$=$	$\frac{3}{4} \cdot 4 + 1$
	↓	shorten
$y$	$=$	$3 + 1$
	↓	add
$y_{S}$	$=$	$4$

Exercises: Intersection of two lines

Calculating the line intersection

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Calculating the line intersection

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Calculating the line intersection

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Calculating the line intersection

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Calculating the line intersection

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Calculating the line intersection

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