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madam [21]
2 years ago
8

Solve the following system of equations by substitution. Show all steps.

Mathematics
1 answer:
Alinara [238K]2 years ago
4 0

Answer:

Given system of equations:

\begin{cases}f(x)=-x^2+2x+3\\g(x)=-2x+3\end{cases}

To solve by substitution, equate the equations and solve for x:

\begin{aligned}f(x) & = g(x)\\\implies -x^2+2x+3 & = -2x+3\\-x^2+4x & = 0\\x^2-4x & = 0\\x(x-4) & = 0\\\implies x & = 0\\\implies x-4 & = 0 \implies x=4\end{aligned}

Therefore, the x-values of the solution are x = 0 and x = 4.

To find the y-values of the solution, substitute the found values of x into the functions:

f(0)=-(0)^2+2(0)+3=3

g(0)=-2(0)+3=3

f(4)=-(4)^2+2(4)+3=-5

g(4)=-2(4)+3=-5

Therefore, the solutions to the given system of equations are:

(0, 3) and (4, -5)

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The diameter is 46 and the radius is 11.5

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Find out if the 3 positions are on the same line or not (3,2,0) (1, -1,2) (5,5, 2)
sattari [20]

Answer:

The three positions are not on the same line.

Step-by-step explanation:

We have three points: A:(3,2,0),  B:(1,-1,2) and C:(5,5,2).

Let's build a vector that goes from one point to another; that vector will be the director of a line (if we draw a vector that goes from A to B, that vector will be the direction of the line that pass by A and B because it does not change). In order to build that vector, let's subtract B from A:

A-B=(3,2,0)-(1,-1,2)=(2,3,-2)

The equation of a line is:

p=p_0+rt

Where p is every singular point of the line, p_0 is a particular point of the line (any that we are sure that it is on the line), r is the director vector and t is the independent variable.

Now we have the director vector: (2,3,-2), and we can verify that A and B are on the line:

p=p_0+(2,3,-2)t

Because there is a value for t that satisfies both A and B: If we do p_0=(3,2,0) and t=0, we are going to obtain the point p=A; and if we do t=-1, we are going to obtain p=B. Let's see that:

p=(3,2,0)+(2,3,-2)*0=(3,2,0)\\p=A\\p=(3,2,0)+(2,3,-2)*-1=(1,-1,2)\\p=B\\

If C is also in the same line, C must accomplish the equation: p=(3,2,0)+(2,3,-2)t, so:

C=(5,5,2)=(3,2,0)+(2,3,-2)t

Let's simplify the equation writing the parametric equation, which is just to write the equation for each dimension:

5=3+2t\\5=2-3t\\2=-2t\\

You can verify that there is not a value of t that satisfies all three equations, so, the point C is not on the same line as A and B; which means that A, B and C are not on the same line.

8 0
2 years ago
Find a point on the ellipsoid x2+y2+4z2=36x2+y2+4z2=36 where the tangent plane is perpendicular to the line with parametric equa
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Answer:

A point on the ellipsoid is (-4,2,2) or (4,-2,-2)

Step-by-step explanation:

Given equation of ellipsoid f(x,y,z) :x^2+y^2+4z^2=36

Parametric equations:

x=-4t-1

y=2t+1

z=8t+3

Finding the gradient of function

\nabla f(x,y,z)=\\\nabla f(x,y,z)=

So, The directions vectors=(-4,2,8)

Now the line is perpendicular to plane when direction vector is parallel to the normal vector of line

\nablaf(x,y,z)=(2x,2y,8z)=\lambda(-4,2,8)

So, 2x=-4\lambda

\Rightarrow x=-2\lambda

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Substitute the value of x , y and z in the ellipsoid equation

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With \lambda = 2

x=-2(2)=-4

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z=2

With\lambda =- 2

x=-2(-2)=4

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Hence a point on the ellipsoid is (-4,2,2) or (4,-2,-2)

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Answer:

19

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