While reviewing before the test, Peter decided to do his homework. He was tasked with writing an equation of a linear function whose graph passes through the points $M= [-2, -1]$ and $N= [3, 3]$.
Peter knew he needed to find the unknown coefficients $a$ and $b$ in the equation $y=ax+b$. Since he was not sure how to start, he plotted both points on a coordinate plane.
Seeing the graph, he realized that he had been taught in class to determine the coefficient $a$ as the ratio of the lengths of the red and blue line segments. So, he wrote: $$ a=\frac54 $$ To find $b$, he decided to substitute the coordinates of point $M$ into the equation of the line: $$ y=\frac54 x+b $$ because $M$ lies on the graph of this linear function. He obtained: $$ \begin{aligned} -1 &=\frac54 (-2)+b \cr -1 &=-\frac52+b \cr b & =\frac32 \end{aligned} $$ Thus, he wrote the equation of the linear function as: $$ y=\frac54 x+\frac32 $$ Peter’s classmates commented on his solution as follows:
Mirek claimed that the lengths of the red and blue line segments are not related to the coefficients "$a$" and "$b$". The only correct procedure is to substitute the coordinates of both points into the equation of the line and solve the resulting system of equations.
Zdena claimed that we could use the lengths of the line segments to determine the coefficient "$a$", but the correct answer should be:
$$
a=\frac45
$$
Monika claimed that the length of the red line segment is equal to the coefficient "$a$" and the length of the blue line segment is equal to the coefficient "$b$". The equation of the line should be: $$ y=5x+4 $$
Martin claimed that Peter correctly found the coefficient "$a$" but made a mistake in substituting the coordinates of $M$. The equation he got should have been: $$ -2=\frac54 (-1)+b $$
Anne fully agreed with Peter’s solution and claimed it was correct.
Who of Peter’s classmates is right?
Zdena
Anne
Mirek
Martin
Monika
Nobody
If we have two points $M=[x_1,y_1]$ and $N=[x_2,y_2]$, where $x_1\neq x_2$, the coefficient "$a$" can be calculated from the formula: $$ a=\frac{y_2-y_1}{x_2-x_1} $$
In our case $M=[-2,-1]$ and $N=[3,3]$, so the difference $y_2-y_1$ equals to $4$ (length of the blue segment) and the difference $x_2-x_1$ equals to $5$ (length of the red segment). In general, we have to be careful because the differences $y_2-y_1$ and $x_2-x_1$ can be negative, while the lengths of the segments cannot be negative.
