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1 year ago

Solve the following system of equations , and find the ordered pair. X =-y-6 and -7x +6y=-10

Mathematics

1 answer:

Blababa [14]1 year ago

5 0

Given:

There are given the equation:

$\begin{gathered} x=-y-6...(1) \\ -7x+6y=-10...(2) \end{gathered}$

Explanation:

From the equation (1):

Put the value of x into the equation (2):

So,

From the equation (2):

$\begin{gathered} \begin{equation*} -7x+6y=-10 \end{equation*} \\ -7(-y-6)+6y=-10 \\ 7y+42+6y=-10 \\ 13y=-10-42 \\ 13y=-52 \\ y=-4 \end{gathered}$

Now,

Put the value of y into the equation (1):

So,

From the first equation:

$\begin{gathered} \begin{equation*} x=-y-6 \end{equation*} \\ x=-(-4)-6 \\ x=4-6 \\ x=-2 \end{gathered}$

Final answer:

Hence, the ordered pair of the solution is shown below:

$(-2,-4)$

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3 years ago

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A piece of cardboard measures 10 ft by 10 ft. Four equal squares of size x are removed from the corners. After removing the squa

Gnesinka [82]

Answer:

The value of x would be $\frac{5}{3}$

Step-by-step explanation:

Given,

The dimension of the cardboard = 10 ft by 10 ft,

∵ After removing four equal squares of size x ( in ft ) from the corners,

The dimension of the resultant box would be,

Length = ( 10 - 2x ) ft,

Width = ( 10 - 2x ) ft,

Height = x ft,

The volume of box,

$V=(10-2x)\times (10 - 2x)\times x=x(10-2x)^2 = x(100 - 40x + 4x^2)=100x - 40x^2 + 4x^3$

Differentiating with respect to x,

$V'=100 - 80x + 12x^2$

Again differentiating with respect to x,

$V''=-80 + 24x$

For maxima or minima,

$V'=0$

$\implies 100 - 80x + 12x^2 = 0$

$\implies 3x^2 - 20x + 25=0$

By quadratic formula,

$x=\frac{20\pm \sqrt{20^2-4\times 3\times 25}}{6}$

$x=\frac{20\pm \sqrt{400 - 300}}{6}$

$x=\frac{20\pm \sqrt{100}}{6}$

$x=\frac{20\pm 10}{6}$

$\implies x = \frac{10}{6}=\frac{5}{3}\text{ or } x = 5$

For x = 5/3, V'' = negative,

While for x = 5, V'' = Positive,

Hence, the value of x would be 5/3 ft for maximising the volume.

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3 years ago

A map has a scale of 1:5000000000 .

erastova [34]

Considering the definition of scale, you obtain:

the real length of the road is 500,000 km.
the real area of the farm is 300 hectares.

<h3>Definition of map</h3>

A map is the conventional graphical representation of a portion of the earth or another celestial body that shows the size and position of elements of the landscape according to the selected scale and projection. That is, a map is a representation of a place, at a size smaller than the actual size.

<h3>Definition of scale</h3>

The scale of a cartographic representation, or map scale (m), is the relationship of similarity between the real dimensions of the geographical space represented and those of its image on the map. In general, it is also defined as the ratio of the lengths of a linear element in the plane and its representation on the terrestrial reference surface.

That is, the scale of the map is defined as the proportionality relationship that exists between a distance measured on the ground and its corresponding measurement on the map.

<h3>Real length of the road</h3>

A map has a scale of 1:5000000000. This indicates that 1 unit on the map is equivalent to 5,000,000,000 in reality.

You know that the road on the map is 10 cm or 0.0001 km long (knowing that 1 km= 100,000 cm).

Then you can apply the following rule of three: if by definition of scale 1 km on the map is equivalent to 5,000,000,000 km in reality, 0.0001 km on the map is equivalent to how much distance in reality?

$real length of the road=\frac{0.0001 kmx 5,000,000,000 km }{1 km}$

<u><em>real length of the road= 500,000 km</em></u>

Finally, the real length of the road is 500,000 km.

You know that the area of a farm of the map is 6 cm square, 0.0006 m sr 0.00000006 hectares (knowing that 1 cm square= 0.0001 m square and 10,000 m square= 1 hectare).

Then you can apply the following rule of three: if by definition of scale 1 hectare on the map is equivalent to 5,000,000,000 hectares in reality, 0.00000006 hectares on the map is equivalent to how much area in reality?

$real area of the farm=\frac{0.00000006 hectaresx 5,000,000,000 hectares }{1 hectare}$