Precalculus

Practice Test

PrecalculusPractice Test

Is the following ordered pair a solution to the system of equations?

1.

$−5x−y=12 x+4y=9 −5x−y=12 x+4y=9$with$(−3,3) (−3,3)$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

2.

$1 2 x− 1 3 y=4 3 2 x−y=0 1 2 x− 1 3 y=4 3 2 x−y=0$

3.

$− 1 2 x−4y=4 2x+16y=2 − 1 2 x−4y=4 2x+16y=2$

4.

$5x−y=1 −10x+2y=−2 5x−y=1 −10x+2y=−2$

5.

$4x−6y−2z= 1 10 x−7y+5z=− 1 4 3x+6y−9z= 6 5 4x−6y−2z= 1 10 x−7y+5z=− 1 4 3x+6y−9z= 6 5$

6.

$x+z=20 x+y+z=20 x+2y+z=10 x+z=20 x+y+z=20 x+2y+z=10$

7.

$5x−4y−3z=0 2x+y+2z=0 x−6y−7z=0 5x−4y−3z=0 2x+y+2z=0 x−6y−7z=0$

8.

$y= x 2 +2x−3 y=x−1 y= x 2 +2x−3 y=x−1$

9.

$y 2 + x 2 =25 y 2 −2 x 2 =1 y 2 + x 2 =25 y 2 −2 x 2 =1$

For the following exercises, graph the following inequalities.

10.

$y< x 2 +9 y< x 2 +9$

11.

$x 2 + y 2 >4 y< x 2 +1 x 2 + y 2 >4 y< x 2 +1$

For the following exercises, write the partial fraction decomposition.

12.

$−8x−30 x 2 +10x+25 −8x−30 x 2 +10x+25$

13.

$13x+2 (3x+1) 2 13x+2 (3x+1) 2$

14.

$x 4 − x 3 +2x−1 x ( x 2 +1) 2 x 4 − x 3 +2x−1 x ( x 2 +1) 2$

For the following exercises, perform the given matrix operations.

15.

$5[ 4 9 −2 3 ]+ 1 2 [ −6 12 4 −8 ] 5[ 4 9 −2 3 ]+ 1 2 [ −6 12 4 −8 ]$

16.

17.

$[ 1 2 1 3 1 4 1 5 ] −1 [ 1 2 1 3 1 4 1 5 ] −1$

18.

$det| 0 0 400 4,000 | det| 0 0 400 4,000 |$

19.

$det| 1 2 − 1 2 0 − 1 2 0 1 2 0 1 2 0 | det| 1 2 − 1 2 0 − 1 2 0 1 2 0 1 2 0 |$

20.

If$det(A)=−6, det(A)=−6,$ what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?

21.

Rewrite the system of linear equations as an augmented matrix.

$14x−2y+13z=140 −2x+3y−6z=−1 x−5y+12z=11 14x−2y+13z=140 −2x+3y−6z=−1 x−5y+12z=11$
22.

Rewrite the augmented matrix as a system of linear equations.

$[ 1 0 3 −2 4 9 −6 1 2 | 12 −5 8 ] [ 1 0 3 −2 4 9 −6 1 2 | 12 −5 8 ]$

For the following exercises, use Gaussian elimination to solve the systems of equations.

23.

$x−6y=4 2x−12y=0 x−6y=4 2x−12y=0$

24.

$2x+y+z=−3 x−2y+3z=6 x−y−z=6 2x+y+z=−3 x−2y+3z=6 x−y−z=6$

For the following exercises, use the inverse of a matrix to solve the systems of equations.

25.

$4x−5y=−50 −x+2y=80 4x−5y=−50 −x+2y=80$

26.

$1 100 x− 3 100 y+ 1 20 z=−49 3 100 x− 7 100 y− 1 100 z=13 9 100 x− 9 100 y− 9 100 z=99 1 100 x− 3 100 y+ 1 20 z=−49 3 100 x− 7 100 y− 1 100 z=13 9 100 x− 9 100 y− 9 100 z=99$

For the following exercises, use Cramer’s Rule to solve the systems of equations.

27.

$200x−300y=2 400x+715y=4 200x−300y=2 400x+715y=4$

28.

$0.1x+0.1y−0.1z=−1.2 0.1x−0.2y+0.4z=−1.2 0.5x−0.3y+0.8z=−5.9 0.1x+0.1y−0.1z=−1.2 0.1x−0.2y+0.4z=−1.2 0.5x−0.3y+0.8z=−5.9$

For the following exercises, solve using a system of linear equations.

29.

A factory producing cell phones has the following cost and revenue functions:$C(x)= x 2 +75x+2,688 C(x)= x 2 +75x+2,688$and$R(x)= x 2 +160x. R(x)= x 2 +160x.$What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

30.

A small fair charges $1.50 for students,$1 for children, and $2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of$1,050. How many of each type of ticket was sold?