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Ber [7]
3 years ago
9

One positive integer is 6 less than twice another. The sum of their squares is 680 . Find the integers.

Mathematics
1 answer:
NemiM [27]3 years ago
4 0

Let's call the two numbers x and y. The first sentence, one is 6 less than twice another, translates to

x = 2y-6

We also know the sum of their squares:

x^2+y^2 = 680

So, we have the following system:

\begin{cases} x = 2y-6 \\ x^2+y^2 = 680 \end{cases}

We can use the expression for x in terms of y from the first equation to turn the second equation into something involving y alone:

x^2+y^2 = 680 \to (2y-6)^2+y^2 = 680

Expand the square:

5y^2-24y+36 = 680

Subtract 680 from both sides:

5y^2-24y-644 = 0

Using the quadratic equation

y_{1,2} = \cfrac{-b\pm\sqrt{b^2-4ac}}{2a}

we find the two solutions

y_1 = -\cfrac{46}{5},\quad y_2 = 14

Since we know that both numbers are integer we can only accept the second solution. It yields the following x value:

x = 2x-6\ \land\ y = 14 \implies x = 2\cdot 14 - 6 = 28-6=22


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3 years ago
If you know how to do it, please help me out :)
Artemon [7]

Answer:

Answer in explanation.

Step-by-step explanation:

Lets take

y =  \frac{3}{4} x - 5

as L1.

Take L2 the line perpendicular to L1.

Standard form of equation of line:

y=mx+c, where m = slope and c = y-intercept.

Since L1 and L2 are perpendicular,

mL1 x mL2 = -1

Substitute mL1 into the equation,

3/4 x mL2 = -1

mL2 = -1 ÷ 3/4

mL2 = -4/3

L2 : y = mx+ c

Substitute y = 6, x = 8 and m = -4/3 into the equation,

6 =  -  \frac{4}{3} (8) + c \\ 6 = -  \frac{32}{3}  + c \\ c = 6 +  \frac{32}{3}  \\  = 16 \frac{2}{3}

therefore L2:

y =  -  \frac{4}{3}x   + 16 \frac{2}{3}

Lets take L3 as the line parallel to L1.

Since L3 and L1 are parallel,

mL3 = mL1 = 3/4

equation of line: y = mx+c

substitute y = 6, x = 8 and m = 3/4 into equation.

6 = ( \frac{3}{4} )(8) + c \\ 6 = 6 + c \\ c = 6 - 6 \\  = 0

therefore L3:

y =  \frac{3}{4} x

8 0
3 years ago
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Ainat [17]

Answer:

The inequality \frac{8}{4}\:  is true.

Step-by-step explanation:

Given

\frac{8}{5}

FALSE

Reason: 8/5 or 1.6 is greater than 9/10 or 0.9  

1>\frac{6}{5}

FALSE

Reason: 1 is smaller than 6/5 or (1.1)

\frac{5}{2}

converting mixed number to improper fraction

\frac{5}{2}

FALSE

Reason: 5/2 can not be less than 5/2

\frac{8}{4}\:

2

TRUE

Reason: 2 is indeed smaller than 5/2 or 5.2        ∵ 5/2 = 2.5

Therefore, inequality \frac{8}{4}\:  is true.

7 0
3 years ago
The length of a rectangle is 4 feet less than 3 times its width. The perimeter of the rectangle is 54 feet. Which equation can b
Mrac [35]

Answer:

B. 2(3w-4)+2w=54

Step-by-step explanation:

L = 3 W - 4

2L + 2W = 54

2 x (3W - 4) + 2W = 54

7 0
4 years ago
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