Help in algebra problem
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Please show us what you have tried and exactly where you are stuck.
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Please share your work/thoughts about this problem.
Dr.Peterson
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As I understand it, you are asking just about this step:
In equation (3), which is [imath]x^2-3x+y=0[/imath], they are replacing x with [imath]\displaystyle\frac{3\lambda+6}{2\lambda+2}[/imath], and y with [imath]\displaystyle3-\frac{\lambda}{2}[/imath]. So they get
[math]\left(\frac{3\lambda+6}{2\lambda+2}\right)^2-3\left(\frac{3\lambda+6}{2\lambda+2}\right)+\left(3-\frac{\lambda}{2}\right)=0[/math]
Then they simplify this.
Is that what you did when you tried replicating their work?
Or maybe you are asking about the last step, how they got x and y from [imath]\lambda[/imath]? This is why you need to show your own thinking, and be specific.
This is a rather abbreviated explanation.
First, do you see where equations 1, 2, and 3 come from?
We end up with three equations in three unknowns. So it is possible for that system to have a finite number of solutions; we just have to find it or them.
Equation 1 has no terms in y. Solve for x in terms of [imath]\lambda_1[/imath]. Equation 2 has no terms in x. Solve for y in terms of [imath]\lambda_1[/imath]. Now substitute those values into equation 3, which gives you a single equation in terms of [imath]\lambda_1[/imath], Solve the resulting cubic using whatever methods you prefer for cubics. Now substitute back to get x and y. It literally is stuff you did in first year algebra.