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Question

# a. Calculate the solution set and b. Demonstrate that your solutions are right by drawing the graphs, preferably by computer. $5 x^2-3 y^2=-22$\$5 x^2-6 y^2+12 y=-85$

Solution

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Step 1
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We have to solve the system

\begin{aligned} 5x^2-3y^2 &= -22\\ 5x^2-6y^2+12y &= -85 \end{aligned}

If we subtract the second equation from the first one, $5x^2$ cancels and we get

\begin{aligned} -3y^2-(-6y^2)-12y &= -22-(-85) && \text{write the equation}\\ 3y^2-12y &= 63 && \text{add }-3y^2+6y^2\text{ and }-22+85\\ y^2-4y &= 21 && \text{divide both sides by }3\\ y^2-4y-21 &= 0 && \text{transfer }21\text{ left}\\ y^2-7y+3y-21 &= 0 && \text{write }-4y\text{ as }-7y+3y\\ y(y-7)+3(y-7) &= 0 && \text{extract }y\text{ and }3\\ (y-7)(y+3) &= 0 && \text{extract }y-7 \end{aligned}

From $y-7=0$ we get a solution $y_1=7$ and from $y+3=0$ we get a solution $y_2=-3$.

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