### Linear formulas and equations: Linear equations and inequalities

### Intersection point of two linear formulas

We have seen how to solve a linear equation. With this 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}#.

We can find the #y#-coordinate by substituting #x=-\tfrac{9}{5}# in one of the formulas. 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}}#.

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.

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 #-7#. The #y#-coordinate of the intersection point is equal to #-1#.

Hence, the intersection point is: #\rv{-7, -1}#.

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