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Section 5.1 Quadratic Systems
Subsection 5.1.1 Solving by Substitution
Substitution is the main technique to solve systems algebraically. Elimination is more awkward.
Exercise Group 5.1.1. Solving Systems by Substitution.
Solve each system of equations.
(a)
\(\begin{cases} y = x^2 - 3x - 4 \\ 2x - y = 4 \end{cases}\)
Answer.
\((0,-4)\text{,}\) \((5,6)\text{.}\)
(b)
\(\begin{cases} x^2 + y = 4 \\ 2x + y = 1 \end{cases}\)
Answer.
\((-1,3)\text{,}\) \((3,-5)\text{.}\)
(c)
\(\begin{cases} 2x^2 - y = 1 \\ y = 5x + 2 \end{cases}\)
Answer.
\(\brac{-\frac{1}{2},-\frac{1}{2}}\text{,}\) \((3,17)\text{.}\)
(d)
\(\begin{cases} x^2 - y = 3 \\ y = 3x + 7 \end{cases}\)
Answer.
\((-2,1)\text{,}\) \((5,22)\text{.}\)
(e)
\(\begin{cases} y = 1 - x^2 \\ x + y = 2 \end{cases}\)
(f)
\(\begin{cases} y = x^2 + 5x + 6 \\ x - y = -12 \end{cases}\)
Answer.
\(\brac{-2+\sqrt{10},10+\sqrt{10}} \approx (1.16,13.16)\text{,}\) \(\brac{-2-\sqrt{10},10-\sqrt{10}} \approx (-5.16,6.84)\text{.}\)
(g)
\(\begin{cases} 2y = x^2 \\ y = x - \frac{1}{2} \end{cases}\)
Answer.
\((1,\frac{1}{2})\text{.}\)
(h)
\(\begin{cases} y = x^2 - 3x - 4 \\ 2x - y = 3 \end{cases}\)
Answer.
\(\brac{\frac{5-\sqrt{29}}{2},2-\sqrt{29}} \approx (-0.19,-3.39)\text{,}\) \(\brac{\frac{5+\sqrt{29}}{2},2+\sqrt{29}} \approx (5.19,7.39)\text{.}\)
(i)
\(\begin{cases} 3x^2 - 10y = 5 \\ x - y = -2 \end{cases}\)
Answer.
\(\brac{-\frac{5}{3},\frac{1}{3}}\text{,}\) \((5,7)\text{.}\)
(j)
\(\begin{cases} 2x^2 - 3y = 2 \\ x - 2y = -2 \end{cases}\)
Answer.
\(\brac{-\frac{5}{4},\frac{3}{8}}\text{,}\) \((2,2)\text{.}\)
(k)
\(\begin{cases} y = \frac{1}{2}x^2 - 2x + 3 \\ 2x - 2y = 3 \end{cases}\)
Answer.
\((3,\frac{3}{2})\text{.}\)
(l)
\(\begin{cases} y = -\frac{1}{2}x^2 + 2x - 3 \\ y = x - 2 \end{cases}\)
(m)
\(\begin{cases} y = x^2 \\ x + y = 3 \end{cases}\)
Answer.
\(\brac{\frac{-1 + \sqrt{13}}{2},\frac{7 - \sqrt{13}}{2}} \approx (1.30,1.70)\text{,}\) \(\brac{\frac{-1-\sqrt{13}}{2},\frac{7 + \sqrt{13}}{2}} \approx (-2.30,5.30)\text{.}\)
(n)
\(\begin{cases} x^2 + 2y = -2 \\ -2x + y = 1 \end{cases}\)
(o)
\(\begin{cases} y = x^2 - x \\ y = 2x \end{cases}\)
Answer.
\((0,0)\text{,}\) \((3,6)\text{.}\)
(p)
\(\begin{cases} y = x^2 - 6x \\ y = x - 12 \end{cases}\)
Answer.
\((3,-9)\text{,}\) \((4,-8)\text{.}\)
(q)
\(\begin{cases} y = x^2 + 8x - 10 \\ y = 3x + 4 \end{cases}\)
Answer.
\((-7,-17)\text{,}\) \((2,10)\text{.}\)
(r)
\(\begin{cases} x^2 = y \\ 2x - y = 1 \end{cases}\)
(s)
\(\begin{cases} x^2 + y = 9 \\ 3x + 2y = 16 \end{cases}\)
Answer.
\(\brac{-\frac{1}{2},\frac{35}{4}}\text{,}\) \((2,5)\text{.}\)
(t)
\(\begin{cases} x^2 - y = 10 \\ 2x - 3y = -10 \end{cases}\)
Answer.
\(\brac{-\frac{10}{3},\frac{10}{9}}\text{,}\) \((4,6)\text{.}\)
(u)
\(\begin{cases} 3y - x - 3 = 0 \\ y + 2x^2 - 2 = 0 \end{cases}\)
Answer.
\(\brac{\frac{-1+\sqrt{73}}{12},\frac{35 + \sqrt{73}}{36}} \approx (0.63,1.21)\text{,}\) \(\brac{\frac{-1-\sqrt{73}}{12},\frac{35 - \sqrt{73}}{36}} \approx (-0.80,0.73)\text{.}\)
(v)
\(\begin{cases} y = x^2 - 4x + 3 \\ y = 3x - 2 \end{cases}\)
Answer.
\(\brac{\frac{7-\sqrt{29}}{2},\frac{17-3\sqrt{29}}{2}} \approx (0.81,0.42)\text{,}\) \(\brac{\frac{7+\sqrt{29}}{2},\frac{17+3\sqrt{29}}{2}} \approx (6.19,16.58)\text{.}\)
(w)
\(\begin{cases} y = -x^2 + 3 \\ 2x + y = 0 \end{cases}\)
Answer.
\((-1,2)\text{,}\) \((3,-6)\text{.}\)
(x)
\(\begin{cases} x^2 + y = 1 \\ 3x - 2y = 4 \end{cases}\)
Answer.
\(\brac{\frac{-3 + \sqrt{57}}{4}, \frac{-25 + 3\sqrt{57}}{8}} \approx (1.14,-0.29)\text{,}\) \(\brac{\frac{-3 - \sqrt{57}}{4}, \frac{-25 - 3\sqrt{57}}{8}} \approx (-2.64,-5.96)\text{.}\)
Exercise Group 5.1.2. Advanced Examples.
Solve each system of equations.
(a)
\(\begin{cases} xy = 12 \\ 2x - y = -2 \end{cases}\)
Answer.
\((-3,-4)\text{,}\)\((2,6)\text{.}\)
(b)
\(\begin{cases} xy = -9 \\ y = -\frac{1}{2}x \end{cases}\)
Answer.
\(\brac{-3\sqrt{2},\frac{3\sqrt{2}}{2}} \approx (-4.24,2.12)\text{,}\) \(\brac{3\sqrt{2},-\frac{3\sqrt{2}}{2}} \approx (4.24,-2.12)\text{.}\)
(c)
\(\begin{cases} xy = 4 \\ y = 2x - 1 \end{cases}\)
Answer.
\(\brac{\frac{1-\sqrt{33}}{4},\frac{-1-\sqrt{33}}{2}} \approx (-1.19,-3.37)\text{,}\) \(\brac{\frac{1+\sqrt{33}}{4},\frac{-1+\sqrt{33}}{2}} \approx (1.69,2.37)\text{.}\)
