Equation solving
„ Example 1
Solve the following equation for x
ay
x
cb
=
s
„ Solution 1:
Multiply both sides by x.
cbx
ay =
s
„ Multiply both sides by s.
ays = cbx
„ Divide both sides by cb and rewite with x on the left.
x=
ays
cb
„ Solution 2:
Cross multiply the terms:
cbx = ays
„ Divide both sides by cb
x=
ays
cb
„ Example 2
Solve the following equation for x
x
2
+
x
3
= 10
„ Multiply each term on both sides by the LCM (least common multiple / pienin yhteinen jaettava)
of the denominators (which is 6 for numbers 2 and 3).
6ÿ
x
+6ÿ
x
2
3
3 x + 2 x = 60
5 x = 60
x = 12
= 6 ÿ 10
„ Example 3
Solve the following equation for x
x-2
2
-
x+3
3
= 10
„ Multiply each term on both sides by the LCM. Write the numerators in brackets.
6 ÿ Hx - 2L
-
6 ÿ Hx + 3L
= 6 ÿ 10
2
3
3 Hx - 2L - 2 Hx + 3L = 60
work it out to find that
x = 72
2
equations_maa1.nb
„ Example 4
Solve the following equation for x
3
x-3
-
x-3
3
=2
»» ÿ 3 ÿ Hx - 3L
„ Multiply each term on both sides by the LCM of the denominators (which is 3·(x – 2) for numbers
3 and x – 2.
3 ÿ Hx - 3L ÿ 3
x-3
3 ÿ Hx - 3L ÿ Hx - 3L
-
3
= 3 ÿ Hx - 3L ÿ 2
„ Do not multiply, cancel instead:
3ÿ1ÿ3
-
1 ÿ Hx - 3L ÿ Hx - 3L
1
1
9 - Hx - 3L ÿ Hx - 3L = 6 Hx - 3L
9 - Ix2 - 6 x + 9M = 6 x - 18
= 3 ÿ Hx - 3L ÿ 2
9 - x2 + 6 x - 9 = 6 x - 18
- x2 + 6 x = 6 x - 18
- x2 = - 18
x2 = 18
x1 =
18 = 3
x2 = -
2
18 = - 3
2
ü Simple Equations
[1]
Solve the following equations for v.
aL d = vt bL t =
d
v
cM a =
v2
2d
dN
v
a
=
b
c
ü Linear equations
Ensimmäisen asteen yhtälöitä
1
The sum of five consecutive integers is 65. Find the largest integer?
2
Find three consecutive (= peräkkäinen) odd integers such that the first plus three times the
second, minus twice the third, is 44.
odd integers are : 2 n + 1, 2 n + 3 ...
equation : H2 n + 1L + 3 H2 n + 3L - 2 H2 n + 5L = 44
n = 11
the integers are 23, 25, 27
check : 23 + 3 * 25 - 2 * 27 = 44
3
Two sides of an isosceles triangle are each 5 cm longer than the base. The perimeter of the
triangle is 100 cm. What are the lengths of the sides and the base?
base = x, sides = x + 5,
perimeter = x + Hx + 5L + Hx + 5L = 100
3 x + 10 = 100
3 x = 100 - 10 = 90
x = 30
base = 30 cm, both sides are 35 cm
4
The second side of a triangle is twice as long as the first side, and the third side is 10 cm longer
than the first side. If the perimeter is 70 cm, what are the lengths of the three sides?
sides are x and 2 x and x + 10
perimeter = x + 2 x + x + 10 = 70
4 x = 60
x = 15
sides are 15 cm, 30 cm and 25 cm
equations_maa1.nb
3
5
The length of a plastic tube is 420 cm. It has to be cut into two parts so that the proportion of
the lengths of the two parts is 2 : 5. Calculate the lenghts of the two parts.
2 x + 5 x = 420
7 x = 420
x = 60 cm
answer : parts are 2 x = 120 cm and 5 x = 300 cm
6
Ground beef costing $2.40 per pound is mixed with ground pork costing $1.80 per pound to
produce 60 pounds of a meatloaf mixture that sells for $1.99 per pound. How many pounds of each
type of meat are used?
7
John took one third of Johannaʼs sweets. Later on he took one fifth of the sweets that were left.
Johanna had now only 16 sweets left. How many sweets did Johanna have in the beginning?
Johanna had x sweets
x
John took first sweets
3
2x
sweets were left
3
1 2x
John took later *
sweets
5
3
x 1 2x
Johanna had x - - *
= 16 sweets left
3 5
3
x = 30 sweets
8
Inkjet printer A costs 2150 mk and the cost of ink is 43 p/page. Printer B costs 2900 mk and
the cost of ink is 30 p/page.
a
Express the cost of printing as the function of pages printed for both printers.
(like y = mx + c)
b
Find the total cost of printing 5000 pages on both printers?
c
After how many pages will printer B be a better buy?
A HxL = 2150 + 0.43 x
B HxL = 2900 + 0.3 x
A H5000L = 4300 mk
B H5000L = 4400 mk
2150 + 0.43 x = 2900 + 0.3 x
-> x º 5800 pages
ANSWER : B is cheaper after about 5800 pages.
Solve@2150 + 0.43 x == 2900 + 0.3 x, xD
88x Ø 5769.23<<
9
Tim subtracts twelve from a number and gets the same result as Elina, who, first added eight to
the same number then divided it by six. What is the number?
x - 12 =
x+8
6
10
A jug of milk when half full weighs 435 g. After some milk has been used, the milk jug is onequarter full and weighs 315 g. How much does the empty jug weigh? What would the jug of milk weigh
if it were full?
11
Two walkers start out from the same place at the same time but in opposite directions and after
2 hours they were 24 km apart. If one was 1 km/h slower than the other, find the speed of each walker
assuming constant walking paces were maintained throughout?
„ Simultaneous Linear Equations
12
An ice cream company sold the the whole stock at discount prices because of a fault in the
refrigerator. One client bought two packs of a special ice cream and three packs of ordinary vanilla ice
cream for 20.75 marks. Another client bought five packs of a special ice cream and four packs of
ordinary vanilla ice cream for 40.50 marks. What were the unit prices for the two kinds of ice cream?
4
12
An ice cream company sold the the whole stock at discount prices because ofequations_maa1.nb
a fault in the
refrigerator. One client bought two packs of a special ice cream and three packs of ordinary vanilla ice
cream for 20.75 marks. Another client bought five packs of a special ice cream and four packs of
ordinary vanilla ice cream for 40.50 marks. What were the unit prices for the two kinds of ice cream?
„ Quadratic Equations
1
Simplify (remove the brackets)
a)
(x – 4)(x + 4) =
b)
(x + 4L2 =
c)
x·(x – 5)(x + 1) =
d)
(5x – 3)(5x + 3) =
2
Solve the following quadratic equations without using the quadratic formula:
a)
x2 – 4 = 0
b)
3x 2 – 27 = 0
c)
(x – 7)(x + 3) = 0
d)
x (x + 7) = 0
3
A rectangle is 4 m longer than it is wide.
What are its dimensions if the area of the rectangle is 6 m2 .
4
One positive number is 6 more than another positive number.
What are the two numbers if their product is 9?
5
One leg of a right triangle is 35 cm shorter than than the other. The hypotenuse is 10 cm
longer than the longer leg. What are the dimensions of the triangle?
6
The horizontal sides of a square are made 3 cm longer and its vertical sides are made 4 cm
shorter. The resulting rectangle has an area of 18 cm2 . Find the length of the side of the original
square.
7
The side of a square is 4 cm longer than the side of a smaller square. the sum of the areas of
the squares is 17 cm2 . What are the lengths of the sides of the squares?
8
A rectangular room is 4 metres wider than it is high and it is 8 metres longer than it is wide.
The total area of the walls is 512 square metres. Find the height of the room.
9
The center one third of a 120-m by 80-m lawn remains to be mowed as shown in the diagram.
What are the dimensions of the uncut portion?
x
80 m
x
120 m
10
The following equations contain fractions. Multiply each term by the common denominator (=
LCM of the denominators)
@aD
@bD
@cD
@dD
3
x
+2x = 7
x+
1
2x
1
x+2
8
2x
+
=
9
5
1
Hx + 2L2
+5 =
x+3
4
=
1
6
equations_maa1.nb
@eD
5
1
x-2
1
-
2
x -4
=
4
5
11
Solving the following simultaneous equations leads also to a quadratic equation. Solve first x
from the first equation and substitute it in the second equation. The solutions are two sets of (x, y)
values.
:
x-6y-5 = 0
xy - 6 = 0
Percentages
1
Convert 0.6 into percentage.
2
Change 40% into a fraction in its lowest terms.
3
Change 35% into a decimal number.
4
Calculate 10% of 980.
5
What percentage is 4 of 25
6
Calculate 75% of 40.
7
25% of a length is 10 cm. What is the complete length?
8
What is 0.25% of £2000?
9
A girl scores 80 marks out of 120 in a test. What was her percentage mark?
10
In an examination 60% of the maximum mark is required for a pass. If the maximum mark is
200, what is the pass mark?
11
In a sale a shopkeeper reduces all his articles by 20%. What is the sale price of an article
originally costing £150?
12
A consignment of perishable goods weighs 300 kg. 30% of the consignment is unsaleable.
What weight is saleable?
„ Answers
1) 60% 2)
2
5
3) 0.35 4) 98 5) 16% 6) 30 7) 40 8) £5 9) 67% 10) 120 11) £120 12) 210 kg
13
A computer consultant earns 8000 mk per month. Taxes take 31% of the income. Rent of the
apartment in 2300 mk per month. She considers a new job where she would earn 10% less but her tax
percentage would be only 30%. How many percent less would her net income be in case she takes
the new job.
ü Direct variation
Two quantities x and y vary directly if there is a constant k, such that
y
x
= k or y = kx
(graph of direct variation is a straight line through the origin)
[if x doubles then also y doubles]
3
2.5
2
1.5
1
0.5
0.5 1 1.5 2 2.5 3
Example:
4 m of material cost 128 mk. How much would 9 m of the same material cost?
Make a table:
cost amount
128
4
ö x = 9 ö x = 288 mk
128
4
x
9
0.5 1 1.5 2 2.5 3
6
Example:
4 m of material cost 128 mk. How much would 9 m of the same material cost?
Make a table:
cost amount
128
4
ö x = 9 ö x = 288 mk
128
4
x
9
equations_maa1.nb
ü Inverse variation
Two quantities x and y vary inversely if there is a constant k, such that x·y=k (graph of inverse variation is a hyperbola)
[if x doubles then y decreases to
1
2
of initial value]
5
4
3
2
1
0.5 1 1.5 2 2.5 3
Example:
distance = speed · time or d = v·t
if v and t are variables and distance is a fixed (constant) number then v and t vary inversely.
Mikko drives 90 km/h from Helsinki to Lahti in 1.2 hours. How long time in minutes does it take for
Maija to cover the same distance at the speed of 110 km/h?
Make a table:
Ñ
speed time HhL
ö 90 * 1.2 = 110 * x ö
Mikko
90
1.2
Maija 110
x
x = 0.982 h º 59 min.
[ if you wish you may write the inverse proportion also as
90
x
= 1.2
110
]
ü Work Problems
1
Five painters will paint the windows of a house in 4 days. How long will it take for two painters
to paint the windows?
painters time
ö 2 x = 20 ö x = 10 days
5
4
2
x
1
room
10
2
Sue paints a room at the rate of
1
room
8
per hour, and Kelly paints the same room at the rate of
per hour. How long will it take them to paint the room together?