Best solutions to - "Proove that knowing that lines y=a1x+b1x2 and y=a2x+b2x2..."

High School
Algebra
Algebra I
Quadratic function and equation

Erik Cantu

Answered question

2022-04-02

Proove that knowing that lines

y = a_{1} x + b_{1} x^{2} and y = a_{2} x + b_{2} x^{2}

coincide in every point we can infer that

a_{1} = a_{2} and b_{1} = b_{2}

Answer & Explanation

Ariel Cantrell

Beginner2022-04-03Added 8 answers

If the $y_{1} = a_{1} x + b_{1} x^{2} and y_{2} = a_{2} x + b_{2} x^{2}$ coincide in every point, this means $y_{1} = y_{2}, \forall x \in R$ .
Let $x \neq 0$ , then we have
$\begin{aligned} a_{1} x + b_{1} x^{2} = a_{2} x + b_{2} x^{2} \\ ⟹ & x^{2} (b_{2} - b_{1}) + x (a_{2} - a_{1}) = 0 \\ ⟹ & x (x (b_{2} - b_{1}) + (a_{2} - a_{1})) = 0 \\ ⟹ & x (b_{2} - b_{1}) + (a_{2} - a_{1}) = 0, \\ ⟹ & x = \frac{a_{1} - a_{2}}{b_{2} - b_{1}}, if b_{1} \neq b_{2} \end{aligned}$
But, that means if $x \neq 0$ , then the coincide is at only one point, which gives a contradiction. Therefore, we deduce that
$b_{1} = b_{2}$
This means, $a_{1} = a_{2}$ .

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Algebra I

Didn't find what you were looking for?