Intersection point of two linear formulas

Linear formulas and equations: Linear equations and inequalities

Intersection point of two linear formulas

We have seen how to solve a linear equation. Using the same technique, we can also determine the coordinates of the intersection points of two linear formulas.

We consider the linear formulas $\blue{f: y =2 \cdot x + 5}$ and $\green{g: y=-3 \cdot x -4}$ . We can find the $x$ -coordinate of the intersection point by solving the equation $2 \cdot x +5=-3 \cdot x-4$ . This is done in the following manner:

$\begin{array}{rcl}2 \cdot x +5&=&-3 \cdot x-4 \\ &&\qquad\blue{\small\text{the equation}} \\ 5 \cdot x +5&=&-4 \\&& \qquad \blue{\small\text{both sides plus }3 \cdot x} \\ 5 \cdot x &=&-9 \\ &&\qquad\blue{\small\text{both sides minus }5}\\x&=&-\dfrac{9}{5}\\ &&\qquad\blue{\small\text{both sides divided by }5} \end{array}$

Hence, the $x$ -coordinate of the intersection point is $x=-\tfrac{9}{5}$ .
In one of the formulas, we can find the $y$ -coordinate by substituting $x=-\tfrac{9}{5}$ . This gives us:

$y=2 \cdot -\tfrac{9}{5}+5=\tfrac{7}{5}$ Hence, the coordinates of the intersection point are $\rv{-\tfrac{9}{5}, \tfrac{7}{5}}$ .

geogebra plaatje

Intersection point of two linear formulas

The $x$ -coordinate of the intersection point of the two linear formulas $y=a \cdot x+b$ and $y=c \cdot x +d$ is the solution to the equation $a \cdot x+b=c \cdot x+d$ . The $y$ -coordinate can be found by substituting the found $x$ -coordinate in one of the linear formulas.

Take a look at the graph below with the linear formulas $f: y=3\cdot x-11$ and $g: y=4\cdot x-13$ .

Find the intersection point in the graph. Give your answer in the form $\rv{a,b}$ , in which $a$ and $b$ are numbers.

The intersection point is: $\rv{2, -5}$ .

After all, in the following graph, the intersection point is drawn in green.

We can see that the $x$ -coordinate of the intersection point is equal to $2$ . The $y$ -coordinate of the intersection point is equal to $-5$ .
Hence, the intersection point is: $\rv{2, -5}$ .

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