Simplification, how?

[whyisthissodifficult]

https://user-images.githubusercontent.com/12116499/71559508-4c626d00-2a5f-11ea-96c8-9054467b0253.jpg contains a simplification on the third line which i don't understand. how a₁x₂ - a₁x₁ gets reduced?

 

[Dr.Peterson]

Factor out a₁ from the LHS, then divide.

 

[whyisthissodifficult]

right, i was trying to follow the "a programmer introduction to mathematics" but haven't touched algebra for almost 2 decades. Could you kindly expand on the steps?

 

[whyisthissodifficult]

oh right, as in a₁(x₂-x₁)=y₂-y₁ => a₁=(y₂-y₁)/(x₂-x₁) ?

 

[JeffM]

oh right, as in a₁(x₂-x₁)=y₂-y₁ => a₁=(y₂-y₁)/(x₂-x₁) ?

Yes

[MATH]a_1x_2 - a_1x_1 = y_2 - y_1 \iff a_1(x_2 - x_1) = y_2 - y_1[/MATH]
This derives trivially from one of the field axioms and is simply an important generalization of

[MATH](2 * 7) - (2 * 3) = 8 = 2(7 - 3).[/MATH]
Therefore

[MATH]a_1x_2 - a_1x_1 = y_2 - y_1 \implies a_1(x_2 - x_1) = y_2 - y_1[/MATH] and

[MATH]a_1(x_2 - x_1) = y_2 - y_1 \implies a_1 = \dfrac{y_2 -y_1}{x_2 - x_1} \text { IF AND ONLY IF } x_2 - x_1 \ne 0.[/MATH]