Intersection points of linear formulas with the axes

Linear formulas and equations: Linear equations and inequalities

Intersection points of linear formulas with the axes

Intersection point with the x-axis

The intersection point with the #x#-axis of the linear formula #y=\blue a \cdot x+\green b# is that point where the #y#-value is equal to #0#.

Therefore, we can find the intersection point of the linear formula #y=\blue a \cdot x +\green b# with the #x#-axis by solving the equation #\blue a \cdot x +\green b =0#.

Hence, the intersection point with the #x#-axis is #\rv{-\frac{\green b}{\blue a},0}#.

geogebra plaatje

Example

Example

Consider the formula #y=-3 \cdot x - 4#.

The intersection point with the #x#-axis can be found by solving #-3 \cdot x-4=0#.

\[\begin{array}{rcl}-3 \cdot x -4 &=& 0\\ && \phantom{xxxxx}\color{blue}{\text{the equation}} \\ -3 \cdot x &=&4 \\&& \phantom{xxxxx}\color{blue}{\text{both sides plus }4} \\x&=&-\frac{4}{3}\\ && \phantom{xxxxx}\color{blue}{\text{both sides divided by}-3} \end{array}\]

Hence, the intersection point with the #x#-axis is: #\rv{-\frac{4}{3},0}#.

Intersection point with the y-axis

The intersection point with the #y#-axis of the linear formula #y=\blue a \cdot x+\green b# is the point where the #x#-value is equal to #0#.

Therefore we can find the intersection point of the linear formula #y=\blue a \cdot x +\green b# with the #y#-axis by substituting #x=0# in the formula.

Hence, the intersection point with the #y#-axis is #\rv{0, \green b}#.

geogebra plaatje

Example

Example

Consider the formula #y=-3 \cdot x - 4#.

The intersection point with the #y#-axis can be found by substituting #x=0# in #y=-3 \cdot x -4#. This gives:

\[y=-3 \cdot 0-4 =-4\]

Hence, the intersection point with the #y#-axis is: #\rv{0,-4}#.

On the line #7 x -8 y = -56# there is point of the #x#-axis and a point of the #y#-axis. The first point has the form #(p,0)# and the second #(0,q)# for certain numbers #p# and #q#. What are #p# and #q#?

#p=-8#
#q=7#

Because if #(p,0)# lies on the line, then #7 p -8\cdot 0 = -56# applies (this follows from entering #x=p# and #y=0# in #7 x -8 y = -56#). This is a linear equation with unknown #p#, where #p=-8# is the solution.

Similarly, entering #x=0# and #y=q# in the equation #7 x -8 y = -56# gives the linear equation #-8\cdot q = -56# with solution #q=7#.

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