Math and Arithmetic

12 and 14

5

Let x and y be the two numbers. 1) x + y = 19

2) xy = 78

Solve (1) for y. Subtract x from both sides.

y = 19 - x

Then we can write equation (2) as x(19-x)=78

Simplifying

x(19 + -1x) = 78

(19 * x + -1x * x) = 78

(19x + -1x2) = 78

Solving

19x + -1x2 = 78

Solving for variable 'x'.

Reorder the terms:

-78 + 19x + -1x2 = 78 + -78

Combine like terms: 78 + -78 = 0

-78 + 19x + -1x2 = 0

Factor a trinomial.

(-13 + x)(6 + -1x) = 0

==

Set the factor '(-13 + x)' equal to zero and attempt to solve:

Simplifying

-13 + x = 0

Solving

-13 + x = 0

Move all terms containing x to the left, all other terms to the right.

Add '13' to each side of the equation.

-13 + 13 + x = 0 + 13

Combine like terms: -13 + 13 = 0

0 + x = 0 + 13

x = 0 + 13

Combine like terms: 0 + 13 = 13

x = 13

Simplifying

x = 13

==

Set the factor '(6 + -1x)' equal to zero and attempt to solve:

Simplifying

6 + -1x = 0

Solving

6 + -1x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-6' to each side of the equation.

6 + -6 + -1x = 0 + -6

Combine like terms: 6 + -6 = 0

0 + -1x = 0 + -6

-1x = 0 + -6

Combine like terms: 0 + -6 = -6

-1x = -6

Divide each side by '-1'.

x = 6

Simplifying

x = 6

==

x = {13, 6} The answer to your question is that the larger number is 13.

10 cm

Perimeter is 100', so each side = 25' and 25 squared = 625. QED

Let the width be x/2 and the length be x:

(x/2)*x = 128

Multiply both sides by 2 and remove the brackets:

x2 = 256

Square root both sides:

x = 16

Therefore: length = 16 feet and width = 8 feet

Check: 16*8 = 128 square feet

16

lenght=15, width= 10

15

14 cm

the answer is 6

EQ1 x+y=8

EQ2 x2+y2=34

EQ1 x=8-y

EQ1 and EQ2 combined gives (8-y)2+y2=34

simplify

64 -16y +y2+y2=34

simplify

2y2-16y+30=0

solve for -16y

(2y-?)(y-!)=0 EQA

EQ3 -2y!-y?=-16y

simplify

EQ3 2!+? =16

EQ4 ?!=30

EQ3 ?=16-2!

Combined EQ3 and EQ4

gives

(16-2!)!=30

simplify

8!-!2=15

Method of exhaustion

8!-!2 when

!=0 -> 0

!=2 ->12

!=3->15

!=3 into EQ3 gives ?=16-2! gives ?=10

substitute into EQA gives (2y-10)(y-3)=0

solving gives y=5 or 3

solving for EQ1 x+y=8 then x=3 or 5

solving for EQ2 x2+y2=34 gives 9+25 for either combination

so the larger number is 5

7 feet

2(7+2) = 18 feet

7*2 = 14 square feet

5

5+4 = 9

52+42 = 41

Let the length be 2x and the width be x:

length*width = area

2x*x = 50

2x2 = 50

x2 = 25

x = 5 feet

A = area, L = length, W = width

A = LW

L = 2W

A = (2W)W = 2W2

2W2 = 50

W = 5 ft

5 - the other number is 3

7