Math 170 Exam Solutions

University:
Santa Ana College
Course:
MATH 170 | Pre-Calculus Mathematics
Academic year:

2023
Views:

192

Pages:

5
Author:

Kundulu8mbj

Math 170 Exam Name: KEY 1) Solve the system algebraically: \[\{\begin{matrix}x^{2}+5y^{2}=5\\ x-y=1\end{matrix}\] \[x^{2}+5y^{2}=5\] \[x-y=1\] \[x=y+1\] \[(y+1)^{2}+5y^{2}=5\] \[y^{2}+2y+1+5y^{2}=5\] \[6y^{2}+2y-4=0\] \[(3y-2)(y+1)=0\] \[y=\frac{2}{3}\] \[y=-1\] \[x=\frac{2}{3}+1\] \[x=\frac{5}{3}\] \[(\frac{5}{3},\frac{2}{3}),(0,-1)\] 2) Solve the system: \(\begin{cases}x+2y+z=6\\ 2x-y+3z=-2\\ x+y-2z=0\end{cases}\) \[x-2y-2x-4y-2z=-12\] \[2x-y+3z=-2\] \[-2x-2y+4z=0\] \[-5y+z=-14\] \[-3y+7z=-2\] \[32y=96\] \[y=3\] \[-3y+7z=-2\] \[-3(3)+7z=-2\] \[7z=7\] \[z=1\] \[x+2(3)+1=6\] \[x+7=6\] \[x=-1\] (1, 3, 1) 5) Find the partial fraction decomposition for: \[\frac{x-26}{15x^{2}-x-6}\] \[\frac{x-26}{(5x+3)(3x-2)}=\frac{A}{5x+3}+\frac{B}{3x-2}\] \[x-26=A(3x-2)+B(5x+3)\] \[=3Ax-2A+5Bx+3B\] \[x-26=(3A+5B)x-2A+3B\] \[\begin{cases}3A+5B=1\\ -2A+3B=-26\end{cases}\Rightarrow\begin{cases}A=7\\ B=-4\end{cases}\] \[\frac{7}{5x+3}-\frac{4}{3x-2}\] 6) Sketch the parabola \(y^{2}=8x\) state the vertex, focus and directrix \[4p=8\] \[p=2\] \[faces right\] 7 focal diameter = 8 \[(2,3)\] 8 → F \[x=-2\] Vertex: (0,0) Focus: (2,0) Directrix: \[X=-2\]

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