EVANS SCHOOL OF PUBLIC POLICY AND GOVERNANCE Math practice problems for incoming MPA students Summer 2015

SOLUTIONS

1. INVERTING EQUATIONS a) 𝑦 = 𝑥 + 4 à 𝒙 = 𝒚 − 𝟒

b) 𝑥 = 2𝑦 + 4 à  𝒚 = 𝟏

𝟐𝒙 − 𝟐

c) 𝑦 = 3𝑥 + 5 à 𝒙 = 𝟏

𝟑𝒚 − 𝟓

𝟑

d) 𝑦 = 4𝑥 −  8 à 𝒙 = 𝟏

𝟒𝒚 + 𝟐

e) 𝑃 = .25𝑄 − 100 à 𝑸 = 𝟒𝑷 + 𝟒𝟎𝟎

f) 𝑦 = − !!𝑥 −  100 à 𝒙 = −𝟒𝒚 −  𝟒𝟎𝟎

g) 𝑃 = −.1𝑄 +  400 à 𝑸 = −𝟏𝟎𝑷 +  𝟒𝟎𝟎𝟎

h) 𝑦 = 32 + !

!𝑥 à 𝒙 = 𝟓

𝟖𝒚 − 𝟐𝟎

2. SOLVING PAIRS OF SIMULTANEOUS EQUATIONS a) 3𝑥 + 𝑦 = 13

𝑥 + 6𝑦 = −  7 𝒚 =  −𝟐, 𝒙 = 𝟓

b) 𝑦 = 𝑥 + 4 𝑥 = 2𝑦 + 4 𝒚 =  −𝟖, 𝒙 = −𝟏𝟐

c) 5𝑎 = 2𝑏 + 3 2𝑎 − 𝑏 = 0 𝒂 =  𝟑,𝒃 = 𝟔

d) 𝑥 + 2𝑦 = 8 𝑥 − 2𝑦 = 4 𝒚 =  𝟏, 𝒙 = 𝟔

e) 4𝑥 + 𝑦 = 9 𝑥 − 𝑦 = 1 𝒚 =  𝟏, 𝒙 = 𝟐

f) 2𝑥 + 3𝑦 =  28 𝑥 + 𝑦 =  11 𝒚 =  𝟔, 𝒙 = 𝟓

g) 11𝑥   +  6𝑦 = 79

11𝑥 + 3𝑦 = 67 𝒚 =  𝟒, 𝒙 = 𝟓

h) !!𝑥   +  !

!𝑦 = 8

23𝑥   +  

32𝑦 = 17

𝒚 =  𝟔, 𝒙 = 𝟏𝟐

3. GRAPHING FROM AN EQUATION OR ITS INVERSE a) Graph 𝑦 = 3𝑥 b) Graph 𝑦 = 𝑥 + 3 c) Graph 𝑦 = !

!𝑥 + 1

d) Graph 𝑦 = !!𝑥 + 8 e) Graph 𝑥 = 𝑦 + 3 f) Graph 𝑥 = !

!𝑦 + 1

g) Graph 𝑦 = 5𝑥 and 𝑦 = −5𝑥 + 50 h) Graph 𝑥 = !

!𝑦 − 7.5 and 𝑦 = − !

!𝑥 + 25

4. RECOVERING THE EQUATION OF A LINE FROM A GRAPH a) Find the equation for the line. b) Find the equation for the line. c) Find the equation for the line.

𝒚 = − 𝟐

𝟑𝒙 + 𝟐𝟏 𝒚 = − 𝟏

𝟏𝟎𝒙 + 𝟗 𝒚 = 𝟑𝒙 + 𝟏𝟎

d) Find the equation for the line. e) Find the equation for the line. f) Find the equation for the line.

𝒚 = − 𝟐𝟓

𝟗𝒙 + 𝟐𝟓 𝒚 = 𝟓

𝟐𝟕𝒙 𝒚 = 𝟏

𝟗𝒙

a) Find the equation for the line. b) Find the equation for the line. c) Find the equation for the line.

𝒚 = − 𝟐

𝟗𝒙 + 𝟐𝟕 𝒚 = 𝟓 𝒙 = 𝟐𝟓

5. CALCULATING AREAS UNDER STRAIGHT LINE CURVES

a) Find the area under the line. b) Find the area under the line. c) Find the area under the line.

(21 X 4) / 2 = 42 (35 X 2) / 2 = 35 (15 X 27) / 2 = 202.5

d) Find the area under the line. e) Find the area under the line. f) Find the area under the line.

((21-6) X 6) / 2 + (6 X 6) = 81 ((40-15) X 9) / 2 + (15 X 9) = 247.5 ((27-6) X 6) / 2 + (6X6) = 99

g) Find the area under the line. h) Find the area under the line. i) Find the area under the line.

(12 X 4) / 2 + ((21-12) X (10-4)) / 2 (30-5)X6 / 2 + (9-6)X5) / 2 ((12-6)X12)/2 + ((15-12) X (24-12)) + (12 X (10-4)) = 123 + (5 X 6) = 112.5 / 2 + (6X12) + (24-12) X 12 = 270

6. NATURAL LOGS AND EXPONENTIAL FUNCTIONS

a) Express ln 2.7183 = 1 in exponential form. à Solution: 2.7183 = e1 b) Write the equation e2.7 ≈ 14.88 in logarithmic form. à Solution: ln 14.88 = 2.7 c) Express the equation e2 ≈ 7.39 in logarithmic form. à Solution: ln 7.39 = 2 d) Write as a single logarithm: ln 3 + ln 7 à Solution: ln 21 (using the product property) e) Write as a single logarithm: ln 6 – ln 2 à Solution: ln 3 (using the quotient property) f) Expand the following expression: ln 12𝑥! à Solution: 𝐥𝐧 𝟏𝟐 + 𝟒 𝐥𝐧 𝒙 (using the product and power

properties) g) Expand the following expression: ln !!

!

!! à Solution: ln 4 + 3 ln y – 5 ln x

h) Solve for x: ln x = 24 à Solution: x = e24 i) Solve for x: e4x+2 = 50 à Solution: x = .478 j) Solve 1 + 2e1-3z = 15. à z = -.315