system 1

\( \begin{cases} 7x-3y=13 \\ x-2y=5 \end{cases} \)

Solve the system

\( \begin{cases} 7x-3y=13 \\ x-2y=5 \end{cases}\),

then \(x=5+2y\)

\(7(5+2y)-3y=13; 35+14y-3y=13; 11y=-22; y=-2\)

\(x=5-4=1\)

\( \begin{cases} x=1 \\ y=-2 \end{cases}\)

proportion 1

\(\frac{x-3}{x-2}=\frac{6,5}{1,5}\)

Solve the equation

\(\frac{x-3}{x-2}=\frac{6,5}{1,5}\)

\(\frac{x-3}{x-2}=\frac{65}{15}=\frac{13}{3}\)

\(3x-9=13x-26\)

\(10x=17; x=1,7\)

task 2

The perimeter of a rectangle is 84 cm. Find the length and the width of the rectangle, if the width refers to the length as 2:5.

The perimeter of a rectangle is 84 cm. Find the length and the width of the rectangle, if the width refers to the length as 2:5.

a - length

b - width

\( \begin{cases} 2a+2b=84 \\ \frac{a}{b}=\frac{2}{5} \end{cases} \)

then \(a=\frac{2b}{5}\)

\(\frac{4b}{5}+2b=84; 4b+10b=420; 14b=420; b=30\)

then \(a=\frac{2\times{30}}{5}=12\)

\( \begin{cases} a=12 \\ b=30 \end{cases} \)

equation 1

\(2x+3-6(x-1)=4(1-x)+5\)

Find the solution

\(2x+3-6(x-1)=4(1-x)+5\)

\(2x+3-6x+6=4-4x+5\)

\(-4x+9=-4x+9\)

\(0=0\)

X is any number.

task 3

For 2 days in the factory there were baked 240 tons of bread. The ratio of the first day baking and the second day baking is 5:3. For how many tons of bread more they baked on the first day than on the second day?

Compute

a - tons of bread baked on the first day. b - tons of bread baked on the second day.

\( \begin{cases} \frac{a}{b}=\frac{5}{3} \\ a+b=240 \end{cases} \)

then \(a=240-b\)

\(\frac{240-b}{b}=\frac{5}{3}\)

\(720-3b=5b; 8b=720; b=90\)

then a=150.

Then 150-90=60. It means that they baked on the 60 tons more on the first day.

equation 2

\(\frac{0,6}{x-3}=\frac{2,1}{x+2}\)

Solve the equation

\(\frac{0,6}{x-3}=\frac{2,1}{x+2}\)

\(\frac{6}{x-3}=\frac{21}{x+2}\)

\(6x+12=21x-63; 15x=75; x=5\)

equation 3

\(6(x-1)=9,4-1,7x\)

Solve the equation

\(6(x-1)=9,4-1,7x; 6x-6=9,4-1,7x; 7,7x=15,4; x=2\)

equation 4

\(\frac{1}{x-3}=2\)

Solve the equation

\(\frac{1}{x-3}=2\)

\(2x-6=1; 2x=7; x=3,5\)

formula 2

Write a formula of numbers that are divisible by 5

Only 5n will always be divisible by 5.

task 9

2 l of the solution containes 10 g of salt. How much salt does 7 l of the solution contain?

2 l of the solution containes 10 g of salt. How much salt does 7 l of the solution contain?

2 l - 10 g of salt;

7 l - x g of salt.

\(\frac{2}{7}=\frac{10}{x}\)

\(x =\frac{70}{2}=35\)

Factorization

\(x(a+b+c)-y(a+b+c)+z(a+b+c)\)

Factorize

\(x(a+b+c)-y(a+b+c)+z(a+b+c)=(a+b+c)(x-y+z)\)

task 10

At the pier there are 6 boats, some of which are doubles, and some are triples. In total, they can accommodate 14 people. How many double and triple boats are at the pier?

Find the solution

"a" - number of doubles, "b" - number of triples.

Then a+b=6; a=6-b

2a+3b=14. 2(6-b)+3b=14; 12+b=14; b=2.

a=4

inequality 1

\(x^2-4x-5\le0\)

\(x^2-4x-5\le0\)

\((x-5)(x+1)\le0\)

\(x\in [-1;5]\)

equation 13

\(\sqrt{5x-1}=\sqrt{3x+1}\)

Solve the equation

\((\sqrt{5x-1})^2=(\sqrt{3x+1})^2\) if \(5x-1>0; 3x+1>0\)

\(5x-1=3x+1\)

\(2x=2\)

\(x=1\)

 

expression 24

\(\frac{5a^2}{a-1}-5a\)

Simplify

\(\frac{5a^2}{a-1}-5a=\frac{5a^2}{a-1}-\frac{5a(a-1)}{a-1}=\frac{5a^2-5a^2+5a}{a-1}=\frac{5a}{a-1}\)

multipliers

\(y^2(2y-5)-8y+20\)

Factorize

\(y^2(2y-5)-8y+20=(2y-5)(y^2-4)=(2y-5)(y-2)(y+2)\)

equation 16

\(\frac{1}{x-3}=2\)

Solve the equation

\(\frac{1}{x-3}=2\)

2(x-3)=1

2x-6=1

2x=7

x=3,5

expression 30

\(\frac{b(a^2-ab+b^2)}{a^3+b^3}+\frac{a}{a+b}\)

Simplify the expression

\(\frac{b(a^2-ab+b^2)}{a^3+b^3}+\frac{a}{a+b}=\frac{b(a^2-ab+b^2)}{(a+b)(a^2-ab+b^2)}+\frac{a}{a+b}=\frac{b}{a+b}+\frac{a}{a+b}=\frac{a+b}{a+b}=1\)

function

\(y=\frac{1}{4+x^2}\)

Find the domain of the function

The denominator of the fraction never equal to 0.

\(x^2\geq0\)

\(4+x^2\geq4\), then it never gets 0. Then x is any number.

task 18

Garden region has the shape of a rectangle, one side of which is 10 meters more than the other. It is required to enclose a fence. Determine the length of the fence, if it is known that the area of the pregion is 1200m2

a  and a+10 - sides

a(a+10)=1200

a2+10a-1200=0

a=-40 (not fit)

a=30

l=2a+2(a+10)=4a+20=120+20=140

expression 36

\(\frac{x-5}{5x-25}-\frac{3x+5}{5x-x^2}\)

Simplify

\(\frac{x-5}{5x-25}-\frac{3x+5}{5x-x^2}=\frac{x-25}{5(x-5)}+\frac{3x+5}{x(x-5)}=\frac{x^2-25x+15x+25}{5x(x-5)}=\frac{x^2-10x+25}{5x(x-5)}=\frac{(x-5)^2}{5x(x-5)}=\frac{x-5}{5x}\)

task 19

There are solutions containing 30% of nitric acid and 55% of nitric acid. How much of each solution one must take to obtain 100 l of 50% nitric acid solution?

x l - the first solution; y l - the second solution

x+y=100; x=100-y

0,3x+0,55y=100*0,5

0,3(100-y)+0,55y=50

30-0,3y+0,55y=50

0,25y=20

y=80; x=20

Sum

Find a when the sum of the fractions \(\frac{2a-1}{4}\) and \(\frac{a-1}{3}\) is positive

Find a

\(\frac{2a-1}{4}+\frac{a-1}{3}=\frac{6a-3+4a-4}{12}=\frac{10a-7}{12}>0\)

10a-7>0

a>0,7

proportion 3

\(1\frac79:x=2\frac23:\frac{3}{100}\)

Find x

\(1\frac79:x=2\frac23:\frac{3}{100}; \frac{16}{9x}=\frac{\frac83}{\frac{3}{100}}; \frac{16}{9x}=\frac{800}{9}; x=\frac{16}{800}=0,02\)

task 21

The boy rode a bicycle from the village to the lake and back, spending 1 hour for the entire trip. From the village to the lake he was driving at a speed of 15 km/h, and back - 10 km/h. What is the distance between the lake and the village?

S - the distance between the lake and the village.

\(\frac{S}{15}+\frac{S}{10}=1\)   |*30

2S+3S=30

5S=30

S=6 km

fraction 3

\(\frac{p^3-125}{p^2+5p+25}\)

Cut the fraction

\(\frac{p^3-125}{p^2+5p+25}=\frac{(p-5)(p^2+5p+25)}{p^2+5p+25}=p-5\)

expresion 24

\(\frac{2-\frac{a-b}{a+b}}{3-\frac{a+2b}{a+b}}\)

Simplify

\(\frac{2-\frac{a-b}{a+b}}{3-\frac{a+2b}{a+b}}=\frac{\frac{2a+2b-a+b}{a+b}}{\frac{3a+3b-a-2b}{a+b}}=\frac{a+3b}{2a+b}\)

task 22

The train passed the distance between the cities in 6 hours, moving at a speed of 70 km/h. How long will it takes for the train to pass this distance, if it will move at a speed of 42 km/h?

70*6=42*x, where x is the time that the train needs to pass the distance at a speed of 42 km/h.

x=420:42=10 h

expresion 26

\(\frac{n^2-m^2}{(\sqrt{n}-\sqrt{m})^2+2\sqrt{mn}}\)

Simplify

\(\frac{n^2-m^2}{(\sqrt{n}-\sqrt{m})^2+2\sqrt{mn}}=\frac{n^2-m^2}{n--2\sqrt{nm}+m+2\sqrt{mn}}=\frac{(n+m)(n-m)}{n+m}=n-m\)

task 23

There are two solutions: the first one containing 30% (by volume) of nitric acid, and the second containing 55% of nitric acid. How much you should take the first and second solutions to obtain 100 liters of a 50% nitric acid solution?

There are two solutions: the first one containing 30% (by volume) of nitric acid, and the second containing 55% of nitric acid. How much you should take the first and second solutions to obtain 100 liters of a 50% nitric acid solution?

"x" - the first solution

"100-x" - the second solution

Then:

0,3x+0,55(100-x)=50

0,3x+55-0,55x=50

0,25x=5

x=20 - the first solution; and 80 l of the second solution.

equation 25

\(2x^2-7x+5=0\)

Solve the equation

\(2x^2-7x+5=0\)

D=49-40=9

\(x = \frac{7+3}{4}\) and \(x = \frac{7-3}{4}\)

x=2,5 and x=1

task 24

From the first to the second pier the boat traveled at a speed of 12 km/h and a half hour after in the same direction the steamer left at 20 km/h. What is the distance between the piers, if steamer arrived 1.5 hours before the boat?

From the first to the second pier the boat traveled at a speed of 12 km/h and a half hour after in the same direction the steamer left at 20 km/h. What is the distance between the piers, if steamer arrived 1.5 hours before the boat?

S - the distance.

\(\frac{S}{12}+0,5+1,5=\frac{S}{20}\)

\(\frac{S+24}{12}=\frac{S}{20}\)

20S+480=12S

8S=480

S=60

task 25

Fresh raspberry contains 85% of water, and dry raspberry - 20% of water. Find the mass of dry raspberries, if fresh was 36 kg.

Fresh raspberry contains 85% of water, and dry raspberry - 20% of water. Find the mass of dry raspberries, if fresh was 36 kg.

Dry substance of the fresh raspberry is 36*(1-0,85)=5,4 kg

Dry rasberry then 5,4:(1-0,2)=6,75 kg

equation 26

\(\frac{2x}{4,8+3,6}=4\frac13\)

Compute

\(\frac{2x}{4,8+3,6}=4\frac13\)

\(\frac{2x}{8,4}=\frac{13}{3}\)

6x=109,2

x=18,2

equation 27

\(25x^2-16=0\)

Compute

\(25x^2=16\)

\(x^2=\frac{16}{25}\)

\(x=\pm\frac45\)

proportion 4

\(15:\frac{1}{20}=x:4\)

Find x

\(\frac{15}{\frac{1}{20}}=\frac{x}{4}\)

\(x=15\times4:\frac{1}{20}\)

x=1200

equation 29

2x-0,1=3x+0,1

Find x

2x-0,1=3x+0,1

3x-2x=-0,1-0,1

x=-0,2

expression 45

\(\frac{28b^6}{c^3}\times\frac{c^5}{84b^6}\)

Simplify

\(\frac{28b^6}{c^3}\times\frac{c^5}{84b^6}=\frac{28c^2}{84}=\frac{c^3}{3}\)

equation 30

Express y in terms of x

10x-5y-7=0

\(y=\frac{10x}{5}-\frac75\)

y=2x-1,4

Expression 46

\(\frac{x^2+4}{(x-2)^3}+\frac{4x}{(2-x)^3}\)

Simplify

\(\frac{x^2+4}{(x-2)^3}+\frac{4x}{(2-x)^3}=\frac{x^2+4-4x}{(x-2)^3}=\frac{(x-2)^2}{(x-2)^3}=\frac{1}{x-2}\)

task 26

There are 420 trees in the park. The number of maples and poplars is proportional to the numbers 3 and 4. For how many more are there poplars than maples?

There are 420 trees in the park. The number of maples and polars is proportional to the numbers 3 and 4. For how many more are there poplars than maples?

Maples: x

Polars: 420-x

\(\frac{x}{420-x}=\frac34\)

4x=1260-3x

7x=1260

x=180

420-x=240

240-180=60

system 3

\(\begin{cases} 3x-2y=9 \\ 6x+2y=36 \\ \end{cases} \)

Solve the system

+\(\begin{cases} 3x-2y=9 \\ 6x+2y=36 \\ \end{cases}\)=> \(\begin{cases} x=5 \\ 15-2y=9 \\ \end{cases}\) => \(\begin{cases} x=5 \\ 2y=6 \\ \end{cases}\) => \(\begin{cases} x=5 \\ y=3 \\ \end{cases}\)

9x=45

x=5