System

Problem 1

Solve the system of equations: $6x - 7y = -8$ and $-x - 4y = -9$ .

See Solution

Problem 2

Solve the system of equations: $10x - 14y = -4$ and $-10x - 20y = -30$ .

See Solution

Problem 3

Solve the system of equations: $5x - 14y = -23$ and $-6x + 7y = 8$ .

See Solution

Problem 4

Lulu had twice as many red buttons as green. After giving away $\frac{1}{3}$ red and $\frac{4}{5}$ green, she had 368 left. How many did she start with?

See Solution

Problem 5

Which expression for $y$ comes from the system: $5x - 8y = 17$ and $-8x - 3y = -43$ ?

See Solution

Problem 6

Find the roller reaction at point A of a symmetric fink roof truss with $s = 3m$ , $h = 1.5m$ , and a vertical load $P = 25kN$ .

See Solution

Problem 7

Analyze how a student could mistakenly choose each incorrect answer for the system of equations: $x - 2y = 2$ and $2x + y = 9$ .

See Solution

Problem 8

Find the intersection point(s) of the demand $q=710-5.5p$ and supply $p=3.5q+57.5$ functions.

See Solution

Problem 9

Find the equilibrium number of workers and hourly wage given the demand $l_{d}=45-5.1 w_{d}$ and supply $l_{s}=-15+4.9 w_{s}$ .

See Solution

Problem 10

Un point $\mathrm{P}$ de masse $\mathrm{m}$ se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour $\mathrm{r}(t)$ et $\omega$ quand $\omega$ est constant.

See Solution

Problem 11

In a charity sale, a pencil box costs \$20 and a pack of chocolate costs \$50. If 100 items are sold for \$3950, how many pencil boxes were sold?

See Solution

Problem 12

Find the intersection points of the parabolas $y=x^{2}+2 x+2$ and $y=-2 x^{2}+8 x-7$ . Choose the correct pairs.

See Solution

Problem 13

Find the number of intersection points for the system: $y=ax+8$ and $y=-x^{2}$ .

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Problem 14

Trova due segmenti la cui somma è $57 \mathrm{~cm}$ e differenza è $7 \mathrm{~cm}$ . R: $32 \mathrm{~cm}$ e $25 \mathrm{~cm}$ .

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Problem 15

36) Trova il lato di un quadrato isoperimetrico a un rettangolo di base $15 \mathrm{~cm}$ e altezza $24 \mathrm{~cm}$ . R: $19,5 \mathrm{~cm}$ 37) Due lati consecutivi di un rettangolo hanno somma $75 \mathrm{~cm}$ e differenza $17 \mathrm{~cm}$ . Trova i lati. R: $46 \mathrm{~cm}, 29 \mathrm{~cm}$

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Problem 16

Правоъгьлен паралелепипед с размери $a, b, c$ има отношение $a:b=4:3$ и $b:c=2:5$ . Сборът на ръбовете е $232 \mathrm{~cm}$ . Какъв е обемът му?

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Problem 17

Giải hệ phương trình: $-2mx + y = 5$ và $mx + 3y = 1$ . Tìm nghiệm khi $m=1$ và giá trị $m$ để hệ có nghiệm duy nhất.

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Problem 18

Solve the system of equations: $x + y = 5$ and $x - y = -1$ .

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Problem 19

Kiaria is 7 years older than Jay, and Martha is twice Kiaria's age. Their ages total 77. Find the age ratio of Jay:Kiaria:Martha.

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Problem 20

Un monedero tiene 125 monedas de 50, 100 y 500, con un total de \$ 17000. ¿Cuántas hay de cada tipo?

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Problem 21

\$ 140 is divided among A, B, and C. A gets \$ 10 more than B, and B gets \$ 20 more than C. Find their amounts.

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Problem 22

Ann has \$200 to buy rulers (\$5 each) and pens (\$6 each).
(i) If she buys 39 items total, how many pens does she buy? (ii) Can she buy 33 items total? Explain.

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Problem 23

Mrs. Lim fried 342 more curry puffs than sardine puffs. After giving away $\frac{1}{3}$ of curry puffs and $\frac{1}{6}$ of sardine puffs, she has 255 left. How many puffs did she give away?

See Solution

Problem 24

Find two natural numbers in the ratio $5:9$ such that $3 \times \text{larger} - 2 \times \text{smaller} = 68$ .

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Problem 25

Two particles $A$ (0.3 kg) and $B$ (0.5 kg) are connected by a string over a pulley. A force of $3.5 \mathrm{~N}$ pulls $B$ down a $30^{\circ}$ incline. Find the tension in the string and the acceleration of $B$ .

See Solution

Problem 26

Find the maximum value of $b$ for the point $(a,b)$ satisfying the inequalities: $y \leq -15x + 3000$ and $y \leq 5x$ .

See Solution

Problem 27

Find the maximum value of $b$ for the point $(a, b)$ that satisfies the inequalities $y \leq -15x + 3000$ and $y \leq 5x$ .

See Solution

Problem 28

Solve for $x$ and $y$ using substitution: 11. $2x - 3y = 5$ , $x = -2y - 8$ .

See Solution

Problem 29

Solve $\frac{x+y}{2x+3y}=\frac{3}{8}$ and $x+2y=225$ for $x$ and $y$ .

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Problem 30

5 pens and 3 rulers cost \$58. If 3 pens cost the same as 4 rulers, find the price of a pen and a ruler.

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Problem 31

Charles has 5 times Denise's stickers. After giving 32 stickers to Denise, they have equal amounts. Find their total stickers.

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Problem 32

Solve the equations $\frac{12}{a} + \frac{6}{b} = 78$ and $\frac{15}{a} + \frac{1}{b} = 78$ .

See Solution

Problem 33

Solve the equations $12x + 6y = 78$ and $15x + y = 78$ , then solve $\frac{12}{a} + \frac{6}{b} = \frac{15}{a} + \frac{1}{b} = 78$ .

See Solution

Problem 34

(a) Solve $12x + 6y = 78$ and $15x + y = 78$ .
(b) Solve $\frac{12}{a} + \frac{6}{b} = 78$ and $\frac{15}{a} + \frac{1}{b} = 78$ .

See Solution

Problem 35

Solve for $a$ and $b$ in the equations: $a + b = 15$ and $a - b = -3$ .

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Problem 36

Given $x=1$ and $y=-1$ , find two equations for constants $a$ and $b$ from $a x + b y = -3$ and $b x - a y = 15$ .

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Problem 37

Given $x=1$ and $y=-1$ , find two equations for constants $a$ and $b$ from $k(1) + b(-1) = -3$ and $b(1) - a(-1) = 15$ . Then, solve for $a$ and $b$ .

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Problem 38

Find the values of $h$ and $k$ where $(12, h)$ is the intersection of $x + 4y = 20$ and $2x + ky - 18 = 0$ .

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Problem 39

Find $k$ and $p$ such that: $p(5k+4) \equiv 0 \pmod{5}$ , $p(7k+5) \equiv 0 \pmod{7}$ , $p(11k+6) \equiv 0 \pmod{11}$ .

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Problem 40

Solve the equations: $3x - y = -6$ and $x + 2y = -23$ to find the numbers $x$ and $y$ .

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Problem 41

Solve for two numbers: $3x - y = 21$ and $x + 2y = 0$ . Find the values of $x$ and $y$ .

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Problem 42

A pet store sold twice as many dogs as cats in March. If total sales in March and April were 500, how many dogs were sold in March? (A) 80 (B) 100 (C) 120 (D) 160 (E) 180

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Problem 43

Solve the system of equations: $-2x = 1$ and $4x - 3y = 0$ .

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Problem 44

Solve the system of equations: $7x + 3y = 0$ and $-5x + 10y = 0$ .

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Problem 45

Solve the system: $x + 3 = -2y + 5$ and $x - 3 = 2y + 7$ . Find $2x$ .

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Problem 46

Prove the system $y' = A y$ is solvable when $A = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ is upper triangular.

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Problem 47

Show that the system $\mathbf{y}^{\prime}=A \mathbf{y}$ can be solved directly for a $2 \times 2$ upper triangular matrix $A$ with constant entries. Use the form $\left[\begin{array}{l} x \\ y \end{array}\right]^{\prime}=\left[\begin{array}{ll} a & b \\ 0 & c \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]$ and solve the second equation first.

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Problem 48

Find the maximum degrees of freedom for a one-component system using the formula $F = C - P + 2$ .

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Problem 49

A fruit stand needs to set prices: $\$ 10$ for 4 apples & 4 oranges, $\$ 12$ for 6 apples & 6 oranges. Unique prices? A) Yes; $\$ 1.00$ apple, $\$ 1.50$ orange. B) Yes; $\$ 1.00$ apple, $\$ 1.00$ orange. C) No; many solutions. D) No; no solution.

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Problem 50

Solve the system of equations: $y=3x-5$ and $x=4y-2$ .

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Problem 51

Solve the system of equations: $x + 4y = 8$ and $x + y = -1$ .

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Problem 52

Solve the system of equations: $x - 2y = 6$ and $x - 2y = -4$ .

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Problem 53

Find the length and width of a rectangular garden with area $96 \mathrm{~m}^{2}$ and perimeter $40 \mathrm{~m}$ .

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Problem 54

Encontre $(x, y)$ com $y>1$ para o sistema: $\log_{y}(9x-35)=6$ e $\log_{3y}(27x-81)=3$ .

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Problem 55

A train station garage has Plan A: R55 + R21/day. Plan B costs for days parked are given. Which system of equations fits?

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Problem 56

Find the maximum of $p=70x+82y$ subject to: $x \geq 0$ , $y \geq 0$ , $x \leq 10$ , $y \leq 20$ , $x+y \geq 5$ , $x+2y \leq 18$ .

See Solution

Problem 57

Solve the system: $2x - 3y = 5$ and $x = -2y - 8$ .

See Solution

Problem 58

Two large and 1 small pump fill a pool in 4 hours. One large and 3 small pumps do the same. How long for 4 large and 4 small pumps?

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Problem 59

Find $P(C)$ given $P(A$ or $C)=\frac{7}{10}$ , $P(A$ and $C)=\frac{2}{5}$ , and $2P(B$ and $C)=P(A$ and $C)$ . Also, find the probability that A, B, or C do NOT occur.

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Problem 60

Solve the system of equations: $2x - 3y = 5$ and $x = -2y - 8$ .

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Problem 61

An adult ticket costs \$5 and a child ticket costs \$2. There were 275 more child tickets sold than adult ones, totaling \$3175. How many adult tickets were sold?

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Problem 62

Mr. Tan paid \$ 385 for 5 rackets and 4 basketballs. Each racket costs \$ 14 more than a basketball. Find racket cost.

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Problem 63

Trei numere naturale $a, b, c$ : $b = 2a + 3$ , $c = 4a$ . Află procentul lui $a$ din $c$ și numerele, suma fiind 87.

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Problem 64

A food truck sells salads for \$6.50 and drinks for \$2.00. If total sales were \$836.50 from 209 items, how many salads?

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Problem 65

Find $x$ if $2^{x} \cdot 3^{y}=18$ and $2^{2x} \cdot 3^{y}=36$ . Options: (a) 2 (b) 1 (c) 3 (d) 4 (e) None.

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Problem 66

Find the value of $x$ given $2^{x+y}=2 \sqrt{2}$ and $2^{x-y}=\sqrt{2}$ . Options: (a) 1 (b) 2 (c) 3 (d) 4 (e) None.

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Problem 67

Find $a$ so the system $x+2y-z=2$ , $2x+y+z=1$ , $-x+4y+a=4$ has no unique solution. Show it has a solution for any value of $a$ .

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Problem 68

Find the value of $5x - y$ given $x + y = 500$ and $3x + 2y = 3000$ .

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Problem 69

1. Acum doi ani, mama avea de 5 ori vârsta fiicei. În 6 ani, fiica va avea 1/3 din vârsta mamei. Poate mama avea 40 de ani? Determină vârsta fiicei.

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Problem 70

Cahaya's chocolate mold has a perimeter of 52 cm. Find the base area using $x+2y=1$ and $\frac{3}{x}-\frac{2}{y}=5$ .

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Problem 71

Bata can produce luxury and standard models using 6000 lbs of plastic and 60 hours of labor. Maximize profit: luxury \$40, standard \$30. Each luxury needs 10 lbs and 8 min labor. How many of each model?

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Problem 72

Solve the system: $x^{2}+y^{2}=25$ and $y=2x$ .

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Problem 73

Rs. 162 is split among $A$ , $B$ , and $C$ with half of $A$ 's, one-third of $B$ 's, and one-fourth of $C$ 's shares equal. Find the shares. (a) $36,54,72$ (b) $72,54,36$ (c) $54,36,72$ (d) $72,36,54$ (e) None of these

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Problem 74

Rs. 366 is shared by $A$ , $B$ , and $C$ where $A = \frac{1}{2}(B+C)$ and $B = \frac{2}{3}(A+C)$ . Find $A$ 's share.

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Problem 75

Find the ratio $a : b : c$ given $a : b = 10 : 7$ and $a : c = 15 : 13$ .

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Problem 76

A purse has 100 nickels and dimes worth \$7. Find the number of each type of coin.

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Problem 77

Dados dos números no nulos con suma, diferencia y producto proporcionais a 5, 3 y 16, halla la suma de sus cifras.

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Problem 78

A 30 gms mix of gold and silver is worth Rs. 1200. If weights are swapped, it's Rs. 1080. Find silver's price per gram.

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Problem 79

A 30g mass of gold and silver is worth Rs. 1200. If exchanged, it's Rs. 1080. Gold is Rs. 50/g. Find silver's price/g.

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Problem 80

A 30 gm mix of gold and silver is worth Rs. 1200. If weights are swapped, it's Rs. 1080. Find silver's price per gram.

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Problem 81

Solve the systems: 1) $2x - 5y = -17$ , $x + 3y = 8$ ; 2) $3u - 4v = -5$ , $-u + 2v = 2$ .

See Solution

Problem 82

Solve the system of equations: $2x + y = 3$ and $3x + 2y = 2$ .

See Solution

Problem 83

Solve the system of equations: $2x + 3y = -1$ and $4x + 7y = 5$ .

See Solution

Problem 84

Solve the system of equations: $2x + 3y = -1$ and $4x + 7y = 5$ .

See Solution

Problem 85

Solve the system of equations: $2x - 3y = 7$ and $4x - 5y = 13$ .

See Solution

Problem 86

Solve the system of equations: $2x - 3y = 7$ and $4x - 5y = 13$ .

See Solution

Problem 87

Solve the following systems of equations:
1) $2x + 3y = -1$ , $4x + 7y = 5$ 2) $2x - 3y = 7$ , $4x - 5y = 13$ 3) $4x - y = 7$ , $2x + 5y = -2$ 4) $3x + 2y = 3$ , $6x + 5y = 9$ 5) $m - n = 4$ , $3m - 10n = 5$

See Solution

Problem 88

Solve the system of equations: $3x + 2y = 3$ and $6x + 5y = 9$ .

See Solution

Problem 89

Solve the system:
$\begin{pmatrix} -5 & 2 \\ 3 & -7 \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 12 \\ -13 \end{pmatrix}$

See Solution

Problem 90

Solve the system:
$\begin{pmatrix} 4 & 3 \\ 9 & 8 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 6 \\ 11 \end{pmatrix}$

See Solution

Problem 91

Solve the system:
$\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$

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Problem 92

Solve the system of equations: $y - 4x = 5$ and $2x - 3y = 6$ .

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Problem 93

There are 356 Grade 8 students. If $34 + 4g = \frac{1}{2}b$ , find the number of boys $b$ and girls $g$ .

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Problem 94

In a factory with 850 workers (400 male), find the male-to-female ratio and how many males were dismissed when the ratio is $4:5$ .

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Problem 95

The ratio of boys to girls is 6:5. After 4 boys and 5 girls leave, the ratio becomes 4:3. Find the original number of girls.

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Problem 96

You have \ $3200 to invest in three funds with returns of$ 10\% $,$ 7\% $, and$ 5\%$. Invest twice as much in growth as in money market. How to allocate?

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Problem 97

Solve the equations: $6x - 3y = 16$ and $9x + 2y = 11$ .

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Problem 98

Solve the equations: 6x - 3y = 16 and 9x + 2y = 11.

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Problem 99

Find $A$ given that $AB=\frac{2}{3}CH$ and $AB+CH=30 \text{ cm}$ .

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Problem 100

In un triangolo rettangolo, i cateti sommano a $21 \mathrm{~cm}$ e differiscono di $3 \mathrm{~cm}$ . Trova l'altezza sull'ipotenusa di $15 \mathrm{~cm}$ .

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System

Problem 1

Solve the system of equations: 6x−7y=−86x - 7y = -86x−7y=−8 and −x−4y=−9-x - 4y = -9−x−4y=−9.

Problem 2

Solve the system of equations: 10x−14y=−410x - 14y = -410x−14y=−4 and −10x−20y=−30-10x - 20y = -30−10x−20y=−30.

Problem 3

Solve the system of equations: 5x−14y=−23 5x - 14y = -23 5x−14y=−23 and −6x+7y=8 -6x + 7y = 8 −6x+7y=8.

Problem 4

Lulu had twice as many red buttons as green. After giving away 13\frac{1}{3}31​ red and 45\frac{4}{5}54​ green, she had 368 left. How many did she start with?

Problem 5

Which expression for yyy comes from the system: 5x−8y=175x - 8y = 175x−8y=17 and −8x−3y=−43-8x - 3y = -43−8x−3y=−43?

Problem 6

Find the roller reaction at point A of a symmetric fink roof truss with s=3ms = 3ms=3m, h=1.5mh = 1.5mh=1.5m, and a vertical load P=25kNP = 25kNP=25kN.

Problem 7

Analyze how a student could mistakenly choose each incorrect answer for the system of equations: x−2y=2x - 2y = 2x−2y=2 and 2x+y=92x + y = 92x+y=9.

Problem 8

Find the intersection point(s) of the demand q=710−5.5pq=710-5.5pq=710−5.5p and supply p=3.5q+57.5p=3.5q+57.5p=3.5q+57.5 functions.

Problem 9

Find the equilibrium number of workers and hourly wage given the demand ld=45−5.1wdl_{d}=45-5.1 w_{d}ld​=45−5.1wd​ and supply ls=−15+4.9wsl_{s}=-15+4.9 w_{s}ls​=−15+4.9ws​.

Problem 10

Un point P\mathrm{P}P de masse m\mathrm{m}m se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour r(t)\mathrm{r}(t)r(t) et ω\omegaω quand ω\omegaω est constant.

Problem 11

In a charity sale, a pencil box costs \$20 and a pack of chocolate costs \$50. If 100 items are sold for \$3950, how many pencil boxes were sold?

Problem 12

Find the intersection points of the parabolas y=x2+2x+2y=x^{2}+2 x+2y=x2+2x+2 and y=−2x2+8x−7y=-2 x^{2}+8 x-7y=−2x2+8x−7. Choose the correct pairs.

Problem 13

Find the number of intersection points for the system: y=ax+8y=ax+8y=ax+8 and y=−x2y=-x^{2}y=−x2.

Problem 14

Trova due segmenti la cui somma è 57 cm57 \mathrm{~cm}57 cm e differenza è 7 cm7 \mathrm{~cm}7 cm. R: 32 cm32 \mathrm{~cm}32 cm e 25 cm25 \mathrm{~cm}25 cm.

Problem 15

Problem 16

Правоъгьлен паралелепипед с размери a,b,ca, b, ca,b,c има отношение a:b=4:3a:b=4:3a:b=4:3 и b:c=2:5b:c=2:5b:c=2:5. Сборът на ръбовете е 232 cm232 \mathrm{~cm}232 cm. Какъв е обемът му?

Problem 17

Giải hệ phương trình: −2mx+y=5-2mx + y = 5−2mx+y=5 và mx+3y=1mx + 3y = 1mx+3y=1. Tìm nghiệm khi m=1m=1m=1 và giá trị mmm để hệ có nghiệm duy nhất.

Problem 18

Solve the system of equations: x+y=5x + y = 5x+y=5 and x−y=−1x - y = -1x−y=−1.

Problem 19

Kiaria is 7 years older than Jay, and Martha is twice Kiaria's age. Their ages total 77. Find the age ratio of Jay:Kiaria:Martha.

Problem 20

Un monedero tiene 125 monedas de 50, 100 y 500, con un total de \$ 17000. ¿Cuántas hay de cada tipo?

Problem 21

\$ 140 is divided among A, B, and C. A gets \$ 10 more than B, and B gets \$ 20 more than C. Find their amounts.

Problem 22

Ann has \$200 to buy rulers (\$5 each) and pens (\$6 each). (i) If she buys 39 items total, how many pens does she buy? (ii) Can she buy 33 items total? Explain.

Problem 23

Mrs. Lim fried 342 more curry puffs than sardine puffs. After giving away 13\frac{1}{3}31​ of curry puffs and 16\frac{1}{6}61​ of sardine puffs, she has 255 left. How many puffs did she give away?

Problem 24

Find two natural numbers in the ratio 5:95:95:9 such that 3×larger−2×smaller=683 \times \text{larger} - 2 \times \text{smaller} = 683×larger−2×smaller=68.

Problem 25

Two particles AAA (0.3 kg) and BBB (0.5 kg) are connected by a string over a pulley. A force of 3.5 N3.5 \mathrm{~N}3.5 N pulls BBB down a 30∘30^{\circ}30∘ incline. Find the tension in the string and the acceleration of BBB.

Problem 26

Find the maximum value of bbb for the point (a,b)(a,b)(a,b) satisfying the inequalities: y≤−15x+3000y \leq -15x + 3000y≤−15x+3000 and y≤5xy \leq 5xy≤5x.

Problem 27

Find the maximum value of bbb for the point (a,b)(a, b)(a,b) that satisfies the inequalities y≤−15x+3000y \leq -15x + 3000y≤−15x+3000 and y≤5xy \leq 5xy≤5x.

Problem 28

Solve for xxx and yyy using substitution: 11. 2x−3y=52x - 3y = 52x−3y=5, x=−2y−8x = -2y - 8x=−2y−8.

Problem 29

Solve x+y2x+3y=38\frac{x+y}{2x+3y}=\frac{3}{8}2x+3yx+y​=83​ and x+2y=225x+2y=225x+2y=225 for xxx and yyy.

Problem 30

5 pens and 3 rulers cost \$58. If 3 pens cost the same as 4 rulers, find the price of a pen and a ruler.

Problem 31

Charles has 5 times Denise's stickers. After giving 32 stickers to Denise, they have equal amounts. Find their total stickers.

Problem 32

Solve the equations 12a+6b=78\frac{12}{a} + \frac{6}{b} = 78a12​+b6​=78 and 15a+1b=78\frac{15}{a} + \frac{1}{b} = 78a15​+b1​=78.

Problem 33

Solve the equations 12x+6y=7812x + 6y = 7812x+6y=78 and 15x+y=7815x + y = 7815x+y=78, then solve 12a+6b=15a+1b=78\frac{12}{a} + \frac{6}{b} = \frac{15}{a} + \frac{1}{b} = 78a12​+b6​=a15​+b1​=78.

Problem 34

(a) Solve 12x+6y=7812x + 6y = 7812x+6y=78 and 15x+y=7815x + y = 7815x+y=78. (b) Solve 12a+6b=78\frac{12}{a} + \frac{6}{b} = 78a12​+b6​=78 and 15a+1b=78\frac{15}{a} + \frac{1}{b} = 78a15​+b1​=78.

Problem 35

Solve for aaa and bbb in the equations: a+b=15a + b = 15a+b=15 and a−b=−3a - b = -3a−b=−3.

Problem 36

Given x=1x=1x=1 and y=−1y=-1y=−1, find two equations for constants aaa and bbb from ax+by=−3a x + b y = -3ax+by=−3 and bx−ay=15b x - a y = 15bx−ay=15.

Problem 37

Given x=1x=1x=1 and y=−1y=-1y=−1, find two equations for constants aaa and bbb from k(1)+b(−1)=−3k(1) + b(-1) = -3k(1)+b(−1)=−3 and b(1)−a(−1)=15b(1) - a(-1) = 15b(1)−a(−1)=15. Then, solve for aaa and bbb.

Problem 38

Find the values of hhh and kkk where (12,h)(12, h)(12,h) is the intersection of x+4y=20x + 4y = 20x+4y=20 and 2x+ky−18=02x + ky - 18 = 02x+ky−18=0.

Problem 39

Find kkk and ppp such that: p(5k+4)≡0(mod5)p(5k+4) \equiv 0 \pmod{5}p(5k+4)≡0(mod5), p(7k+5)≡0(mod7)p(7k+5) \equiv 0 \pmod{7}p(7k+5)≡0(mod7), p(11k+6)≡0(mod11)p(11k+6) \equiv 0 \pmod{11}p(11k+6)≡0(mod11).

Problem 40

Solve the equations: 3x−y=−63x - y = -63x−y=−6 and x+2y=−23x + 2y = -23x+2y=−23 to find the numbers xxx and yyy.

Solve the system of equations: $6x - 7y = -8$ and $-x - 4y = -9$ .

Solve the system of equations: $10x - 14y = -4$ and $-10x - 20y = -30$ .

Solve the system of equations: $5x - 14y = -23$ and $-6x + 7y = 8$ .

Lulu had twice as many red buttons as green. After giving away $\frac{1}{3}$ red and $\frac{4}{5}$ green, she had 368 left. How many did she start with?

Which expression for $y$ comes from the system: $5x - 8y = 17$ and $-8x - 3y = -43$ ?

Find the roller reaction at point A of a symmetric fink roof truss with $s = 3m$ , $h = 1.5m$ , and a vertical load $P = 25kN$ .

Analyze how a student could mistakenly choose each incorrect answer for the system of equations: $x - 2y = 2$ and $2x + y = 9$ .

Find the intersection point(s) of the demand $q=710-5.5p$ and supply $p=3.5q+57.5$ functions.

Find the equilibrium number of workers and hourly wage given the demand $l_{d}=45-5.1 w_{d}$ and supply $l_{s}=-15+4.9 w_{s}$ .

Un point $\mathrm{P}$ de masse $\mathrm{m}$ se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour $\mathrm{r}(t)$ et $\omega$ quand $\omega$ est constant.

Find the intersection points of the parabolas $y=x^{2}+2 x+2$ and $y=-2 x^{2}+8 x-7$ . Choose the correct pairs.

Find the number of intersection points for the system: $y=ax+8$ and $y=-x^{2}$ .

Trova due segmenti la cui somma è $57 \mathrm{~cm}$ e differenza è $7 \mathrm{~cm}$ . R: $32 \mathrm{~cm}$ e $25 \mathrm{~cm}$ .

Правоъгьлен паралелепипед с размери $a, b, c$ има отношение $a:b=4:3$ и $b:c=2:5$ . Сборът на ръбовете е $232 \mathrm{~cm}$ . Какъв е обемът му?

Giải hệ phương trình: $-2mx + y = 5$ và $mx + 3y = 1$ . Tìm nghiệm khi $m=1$ và giá trị $m$ để hệ có nghiệm duy nhất.

Solve the system of equations: $x + y = 5$ and $x - y = -1$ .

Ann has \$200 to buy rulers (\$5 each) and pens (\$6 each).
(i) If she buys 39 items total, how many pens does she buy? (ii) Can she buy 33 items total? Explain.

Mrs. Lim fried 342 more curry puffs than sardine puffs. After giving away $\frac{1}{3}$ of curry puffs and $\frac{1}{6}$ of sardine puffs, she has 255 left. How many puffs did she give away?

Find two natural numbers in the ratio $5:9$ such that $3 \times \text{larger} - 2 \times \text{smaller} = 68$ .

Two particles $A$ (0.3 kg) and $B$ (0.5 kg) are connected by a string over a pulley. A force of $3.5 \mathrm{~N}$ pulls $B$ down a $30^{\circ}$ incline. Find the tension in the string and the acceleration of $B$ .

Find the maximum value of $b$ for the point $(a,b)$ satisfying the inequalities: $y \leq -15x + 3000$ and $y \leq 5x$ .

Find the maximum value of $b$ for the point $(a, b)$ that satisfies the inequalities $y \leq -15x + 3000$ and $y \leq 5x$ .

Solve for $x$ and $y$ using substitution: 11. $2x - 3y = 5$ , $x = -2y - 8$ .

Solve $\frac{x+y}{2x+3y}=\frac{3}{8}$ and $x+2y=225$ for $x$ and $y$ .

Solve the equations $\frac{12}{a} + \frac{6}{b} = 78$ and $\frac{15}{a} + \frac{1}{b} = 78$ .

Solve the equations $12x + 6y = 78$ and $15x + y = 78$ , then solve $\frac{12}{a} + \frac{6}{b} = \frac{15}{a} + \frac{1}{b} = 78$ .

(a) Solve $12x + 6y = 78$ and $15x + y = 78$ .
(b) Solve $\frac{12}{a} + \frac{6}{b} = 78$ and $\frac{15}{a} + \frac{1}{b} = 78$ .

Solve for $a$ and $b$ in the equations: $a + b = 15$ and $a - b = -3$ .

Given $x=1$ and $y=-1$ , find two equations for constants $a$ and $b$ from $a x + b y = -3$ and $b x - a y = 15$ .

Given $x=1$ and $y=-1$ , find two equations for constants $a$ and $b$ from $k(1) + b(-1) = -3$ and $b(1) - a(-1) = 15$ . Then, solve for $a$ and $b$ .

Find the values of $h$ and $k$ where $(12, h)$ is the intersection of $x + 4y = 20$ and $2x + ky - 18 = 0$ .

Find $k$ and $p$ such that: $p(5k+4) \equiv 0 \pmod{5}$ , $p(7k+5) \equiv 0 \pmod{7}$ , $p(11k+6) \equiv 0 \pmod{11}$ .

Solve the equations: $3x - y = -6$ and $x + 2y = -23$ to find the numbers $x$ and $y$ .

Solve for two numbers: $3x - y = 21$ and $x + 2y = 0$ . Find the values of $x$ and $y$ .

Solve the system of equations: $-2x = 1$ and $4x - 3y = 0$ .

Solve the system of equations: $7x + 3y = 0$ and $-5x + 10y = 0$ .

Solve the system: $x + 3 = -2y + 5$ and $x - 3 = 2y + 7$ . Find $2x$ .

Prove the system $y' = A y$ is solvable when $A = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ is upper triangular.

Find the maximum degrees of freedom for a one-component system using the formula $F = C - P + 2$ .

A fruit stand needs to set prices: $\$ 10$ for 4 apples & 4 oranges, $\$ 12$ for 6 apples & 6 oranges. Unique prices? A) Yes; $\$ 1.00$ apple, $\$ 1.50$ orange. B) Yes; $\$ 1.00$ apple, $\$ 1.00$ orange. C) No; many solutions. D) No; no solution.

Solve the system of equations: $y=3x-5$ and $x=4y-2$ .

Solve the system of equations: $x + 4y = 8$ and $x + y = -1$ .

Solve the system of equations: $x - 2y = 6$ and $x - 2y = -4$ .

Find the length and width of a rectangular garden with area $96 \mathrm{~m}^{2}$ and perimeter $40 \mathrm{~m}$ .

Encontre $(x, y)$ com $y>1$ para o sistema: $\log_{y}(9x-35)=6$ e $\log_{3y}(27x-81)=3$ .

Find the maximum of $p=70x+82y$ subject to: $x \geq 0$ , $y \geq 0$ , $x \leq 10$ , $y \leq 20$ , $x+y \geq 5$ , $x+2y \leq 18$ .

Solve the system: $2x - 3y = 5$ and $x = -2y - 8$ .

Find $P(C)$ given $P(A$ or $C)=\frac{7}{10}$ , $P(A$ and $C)=\frac{2}{5}$ , and $2P(B$ and $C)=P(A$ and $C)$ . Also, find the probability that A, B, or C do NOT occur.

Solve the system of equations: $2x - 3y = 5$ and $x = -2y - 8$ .

Trei numere naturale $a, b, c$ : $b = 2a + 3$ , $c = 4a$ . Află procentul lui $a$ din $c$ și numerele, suma fiind 87.

Find $x$ if $2^{x} \cdot 3^{y}=18$ and $2^{2x} \cdot 3^{y}=36$ . Options: (a) 2 (b) 1 (c) 3 (d) 4 (e) None.

Find the value of $x$ given $2^{x+y}=2 \sqrt{2}$ and $2^{x-y}=\sqrt{2}$ . Options: (a) 1 (b) 2 (c) 3 (d) 4 (e) None.

Find $a$ so the system $x+2y-z=2$ , $2x+y+z=1$ , $-x+4y+a=4$ has no unique solution. Show it has a solution for any value of $a$ .

Find the value of $5x - y$ given $x + y = 500$ and $3x + 2y = 3000$ .

Cahaya's chocolate mold has a perimeter of 52 cm. Find the base area using $x+2y=1$ and $\frac{3}{x}-\frac{2}{y}=5$ .

Solve the system: $x^{2}+y^{2}=25$ and $y=2x$ .

Rs. 162 is split among $A$ , $B$ , and $C$ with half of $A$ 's, one-third of $B$ 's, and one-fourth of $C$ 's shares equal. Find the shares. (a) $36,54,72$ (b) $72,54,36$ (c) $54,36,72$ (d) $72,36,54$ (e) None of these

Rs. 366 is shared by $A$ , $B$ , and $C$ where $A = \frac{1}{2}(B+C)$ and $B = \frac{2}{3}(A+C)$ . Find $A$ 's share.