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Chapter 13: Problem 40

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}=5 \\ 2 x-y=3 \end{array}\right.$$

Short Answer

Expert verified
The solutions are (1, -1) and \((\frac{1}{5}, \frac{-13}{5})\)

Step by step solution

01

Simplify the First Equation

Rewrite the first equation from its circle equation form to a more standard form. The equation \((x-1)^{2}+(y+1)^{2}=5\) can be expanded and simplified to \(x^2-2x+y^2+2y=4\)
02

Solve Linear Equation for y

We can rewrite the linear equation for y in terms of x. The equation \(2x - y = 3\) becomes \(y = 2x - 3\)
03

Substitute y in the Expanded First Equation

Substitute the expression for y from step 2 into the expanded first equation from step 1. This yields \(x^2 - 2x + (2x-3)^2 + 2(2x-3) = 4\). This simplifies to a quadratic equation in terms of x: \(5x^2 - 8x + 1 = 0\)
04

Solve for x

At this stage we have a quadratic equation. Let's solve it using the quadratic formula \(x = \frac{-b卤\sqrt{b^2-4ac}}{2a}\). Here a=5, b=-8, and c=1. This gives us two solutions: \(x_1 = 1\) and \(x_2 = \frac{1}{5}\)
05

Find Corresponding y for each x

Now we find the y-coordinate for each solution for x in step 4 by substituting \(x_1\) and \(x_2\) back into the rearranged equation from step 2. This yields two pairs of solutions: when \(x_1 = 1\), \(y_1 = -1\) and when \(x_2 = \frac{1}{5}\), \(y_2 = -\frac{13}{5}\)

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Find the slope of the line passing through \((-2,-3)\) and \((1,5)\).

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