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Equation of line one is $y=a(\alpha-\beta)(x-\alpha)$.

Equation of line two is $y=a(\beta-\alpha)(x-\beta)$.

We have to find the point of intersection of these lines which is $\alpha+\beta$ by 2 as given in the question.

Let me know if any information is missing!

What I tried to do : Equated both equations since they are equal to y but I did not get the answer, I maybe made a mistake while distributing.

ATS
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1 Answers1

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The point of intersection of two lines is, by definition, the point at which the equations of both lines have the same X and Y values. The general method of solving these types of problems is:

  1. Solve each equation for the same variable (i.e. isolate the same indeterminate quantity).

(You already have your equations solved for Y, so that step is done.)

  1. Set the equations equal to each other.

(Since the equality operator is transitive, if Y is equal to two separate things, then those two separate things must also be equal to each other.)

  1. Perform algebra to isolate the unknown variables.

(You can perform any operation to either side of the equality, and the equality will remain true.)

Once you have your X-variable isolated (i.e. alone on one side of the equals sign), you are done. (This is true in determined systems of equations of two variables only. If you had more than two variables, you would have to perform this process for each variable in the form of Gaussian elimination.)

Toasted Uranium
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  • Can you explain why this is as such? I understand that the coordinates of the point will satisfy both equations but i don't understand why they would be equal? – ATS Nov 11 '23 at 20:31
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    The fundamental question is what a linear equation is meant to represent. In a linear equation, if you distribute all the variables so that you have every multiple of X in one term and every constant in another term, then you can see more easily what linear equations represent. In such an equation, the constant is the value that the line will take (i.e. the value Y will take) when the independent (input) variable is zero. The coefficient of X represents the amount by which every increase in the X value will increase the Y value. This is another way of describing the slope, or steepness. – Toasted Uranium Nov 11 '23 at 20:35
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    In equations of the form $y = mx + b$, $m$ is the slope and $b$ is that constant value (also called the y-intercept, since it is the value at which the line intersects the y-axis). – Toasted Uranium Nov 11 '23 at 20:36
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    Equations of this form, when we set $m$ and $b$ to some constant value while allowing $x$ to vary and $y$ to be calculated by the values of the other three variables, can uniquely describe any line. – Toasted Uranium Nov 11 '23 at 20:37
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    Then, if two such lines intersect at only one point, it must be the case that the point at which they intersect has unique $x$ and $y$ values, and that these values are the same for both equations in question. – Toasted Uranium Nov 11 '23 at 20:38
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    At that point, you set either $y$ or $x$ from one equation to equal the same from the other equation, since you already know that they must be the same based on the fact that, at the point you're looking for, they are the same by definition. – Toasted Uranium Nov 11 '23 at 20:39
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    If $y = mx + b$ and $y = nx + c$, then $mx + b = cx + d$, and $mx - cx = d - b$, and $x(m-c) = d - b$, and $x = \frac{d-b}{m-c}$. – Toasted Uranium Nov 11 '23 at 20:43
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    $m$, $n$, $b$, and $c$ can be replaced with any constant value expression, and as long as you can get two linear equations into the form I described, you can find the x-values at which they are equal. – Toasted Uranium Nov 11 '23 at 20:44
  • No problem! I enjoy being absurdly specific as well as making no assumptions whatsoever. – Toasted Uranium Nov 11 '23 at 20:48