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3 years ago

Help please I don’t understand this math problem

Mathematics

1 answer:

Semenov [28]3 years ago

4 0

If you divide (6x +3) by (x +1) you get some quotient and some remainder. You can do it a variety of ways, including synthetic division and long division. The method used here is to rewrite 6x+3 as a multiple of x+1 with some constant term added.
.. 6x +3 = (6x +3) +3 -3
.. = (6x +3 +3) -3
.. = (6x +6) -3
.. = 6(x +1) -3

Now, you can divide this by (x +1) and you have
$\frac{6x+3}{x+1} = \frac{6(x+1)-3}{x+1} = \frac{6(x+1)}{x+1} +\frac{-3}{x+1} = \frac{-3}{x+1}+6$
Then the boxes can be filled from ...
$g(x)=\frac{6(x+1)+(-3)}{x+1}=\frac{-3}{x+1}+6$

You know that
.. f(x) +6
represents a translation of f(x) by 6 units up
And you know that
.. f(x +1)
represents a translation of f(x) by 1 unit left

So, you can figure that
.. g(x) = f(x +1) +6
will represent a translation of 1 unit left and 6 units up of f(x) = -3/x.

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What is the solution to the system of linear equations? −9x+4y=55 −11x+7y=82

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Here is my process with mathematical expression interpreter, LaTeX. Full process given below to obtain the values of variable "x" and "y" altogether, solutions to these system of linear equations.

$\begin{bmatrix}-9x & + & 4y & = & 55 & \bf{- - - \: Eq. \: 1} \\ \\ -11x & + & 7y & = & 82 & \bf{- - - \: Eq. \: 2} \end{bmatrix}$

Now here, we should isolate a variable, or take it as a separate form to find the equation, and furthermore substitute the value of variable "y" into the original isolation of "x", to obtain both the solutions for this linear system of equation. Perform this on equation number 1 (Eq. 1).

Subtract the variable attached value by "4y" on both the sides, in current expression.

$\mathbf{-9x + 4y - 4y = 55 - 4y}$

$\mathbf{-9x = 55 - 4y}$

Both the sides, perform a division of value "-9".

$\mathbf{\dfrac{-9x}{-9} = \dfrac{55}{-9} - \dfrac{4y}{-9}}$

$\mathbf{x = \dfrac{55 - 4y}{-9}}$

$\mathbf{x = - \dfrac{55 - 4y}{9}}$

Substitute or just plug the value of newly obtained expression for variable "x" into Equation, numbered as "2" (Eq. 2.) and isolate further for the variable "y", to obtain first solution for this linear equation.

$\mathbf{-11 \Bigg(- \dfrac{55 - 4y}{9} \Bigg) + 7y = 82}$

$\mathbf{\dfrac{(55 - 4y) \times 11}{9} + 7y = 82}$

Multiply both the sides by a value of "9".

$\mathbf{\dfrac{(55 - 4y) \times 11}{9} + 7y \times 9 = 82 \times 9}$

$\mathbf{11 (55 - 4y) + 63y = 738}$

$\mathbf{605 - 44y + 63y = 738}$

$\mathbf{605 + 19y = 738}$

Subtract both the sides by a value of "- 605".

$\mathbf{605 + 19y - 605 = 738 - 605}$

$\mathbf{19y = 133}$

Divide both the sides by "19".

$\mathbf{\dfrac{19y}{19} = \dfrac{133}{19}}$

$\boxed{\mathbf{y = 7}}$

Substitute this variable value of "y = 7" , into our original isolation for variable "x", the expression is to be substituted by that value to complete the solutions for the linear equations. That is:

$\mathbf{x = - \dfrac{55 - 4y}{9}; \quad y = 2}$

$\mathbf{\therefore \quad x = - \dfrac{55 - 4 \times 7}{9}}$

$\mathbf{\therefore \quad x = - \dfrac{55 - 28}{9}}$

$\mathbf{x = - \dfrac{27}{9}}$

$\boxed{\mathbf{x = - 3}}$

Finalised solutions for these linear system of equations for two components , is:

$\boxed{\mathbf{\underline{\therefore \quad Final \: Solutions \: for \: these \: System \: of \: Linear \: Equations: \: x = - 3, \: \: y = 7}}}$

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