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Lana71 [14]
3 years ago
5

Find all solutions by transforming the system to reduced echelon form and back substituting. (If there are an infinite number of

solutions use s1 as your parameter. If there is no solution, enter NO SOLUTION.)
2x1 + x2 = 2

−x1 − x2 − x3 = 4
Mathematics
1 answer:
Zolol [24]3 years ago
3 0

Answer:

The system has infinite solutions described in the set \{(x_1,x_2,x_3)=(6+s1, -10-2s1,s1): s1\in\mathbb{R}\}

Step-by-step explanation:

The augmented matrix of the system is \left[\begin{array}{cccc}-1&-1&-1&4\\2&1&0&2\end{array}\right].

We apply row operations:

1. We add the first row to the second row twice and obtain the matrix \left[\begin{array}{cccc}-1&-1&-1&4\\0&-1&-2&10\end{array}\right]

2. multiply by -1 the rows of the previous matrix and obtain the matrix \left[\begin{array}{cccc}1&1&1&-4\\0&1&2&-10\end{array}\right] that is the reduced echelon form of the matrix associated to the system.

Now we aply backward substitution:

1. Observe that the reduced echelon form has a free variable, then the system has infinite solutions.

2.

x_2+2x_3=-10\\x_2=-10-2x_3

3.

x_1+x_2+x_3=-4\\x_1-10-2x_3+x_3=-4\\x_1-x_3=-4+10\\x_1=6+x_3.

Then the set of solutions is \{(x_1,x_2,x_3)=(6+s1, -10-2s1,s1): s1\in\mathbb{R}\}

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A standard form of a parabola with points through (2, 0) (3, 2) (4, 6)
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The standard form of a parabola with points through (2, 0) (3, 2) (4, 6)

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Step-by-step explanation:

The standard form of a parabola is y = ax² + bx + c, where a, b , c are constant

To find a , b , c

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∵ The standard form of a parabola is y = ax² + bx + c

∵ The parabola passes through points (2 , 0) , (3 , 2) , (4 , 6)

- Substitute the coordinates of each point in the equation

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∵ x = 2 and y = 0

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- Switch the two sides

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Point (3 , 2)

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- Switch the two sides

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∵ x = 4 and y = 6

∴ 6 = a(4)² + b(4) + c

∴ 6 = 16a + 4b + c

- Switch the two sides

∴ 16a + 4b + c = 6 ⇒ (3)

Subtract equation (1) from equations (2) and (3)

∴ 5a + b = 2 ⇒ (4)

∴ 12a + 2b = 6 ⇒ (5)

- Multiply equation (4) by -2 to eliminate b

∴ -10a - 2b = -4 ⇒ (6)

- Add equations (5) and (6)

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- Add like terms

∴ -2 + c = 0

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∵ y = ax² + bx + c

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∴ y = (1)x² + (-3)x + (2)

∴ y = x² - 3x + 2

The standard form of a parabola with points through (2, 0) (3, 2)

(4, 6) is y = x² - 3x + 2

There is another solution you can substitute the x-coordinate of each point in each answer to find the corresponding value of y, if the value of y gives the same value of the y-coordinate of the point for the three points then this answer is the standard form of the parabola

Learn more:

You can learn more about the parabola in brainly.com/question/8054589

#LearnwithBrainly

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