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3 years ago

Simplify the algebraic expression by combining like terms. Using algebra tiles might be helpful.

Mathematics

2 answers:

NeX [460]3 years ago

6 0

Answer:

8x + 4

Step-by-step explanation:

Simplified

group like terms: (+5x + 1X + 2x) = 8x (-4 + 8) = 4 add the positive sign so you can get an equation: +4, so your simplified expression would be 8x + 4

Sladkaya [172]3 years ago

4 0

Simplify all like terms:

8x + 4

Hope this helps!!

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Q2) If z is directly proportional to x and z = 12 when x = 3, find the value of x when z = 18.

LenKa [72]

Answer:

4.5

Step-by-step explanation:

→ Set up the direct proportion equation

z = kx

→ Substitute in the values

12 = 3k

→ Divide both sides by 3 to isolate k

4 = k

→ Substitute the value of k back into the original direct proportion equation

z = 4x

→ Substitute the value of z in

18 = 4x

→ Divide both sides by 4 to isolate x

4.5 = x

4 0

4 years ago

Read 2 more answers

Factor completely, then place the answer in the proper location on the grid. <br> 6х^2 - 3х - 30

Tems11 [23]

Answer:

3(x + 2)(2x - 5)

Step-by-step explanation:

Given

6x² - 3x - 30 ← factor out 3 from each term

= 3(2x² - x - 10) ← factor the quadratic

Consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term

product = 2 × - 10 = - 20 and sum = - 1

The factors are + 4 and - 5

Use these factors to split the x- term

2x² + 4x - 5x - 10 ( factor the first/second and third/fourth terms )

= 2x(x + 2) - 5(x + 2) ← factor out (x + 2) from each term

= (x + 2)(2x - 5), thus

2x² - x - 10 = (x + 2)(2x - 5) and

6x² - 3x - 30

= 3(x + 2)(2x - 5) ← in factored form

7 0

3 years ago

Name/ Uid:1. In this problem, try to write the equations of the given surface in the specified coordinates.(a) Write an equation

Gemiola [76]

To find:

(a) Equation for the sphere of radius 5 centered at the origin in cylindrical coordinates

(b) Equation for a cylinder of radius 1 centered at the origin and running parallel to the z-axis in spherical coordinates

Solution:

(a) The equation of a sphere with center at (a, b, c) & having a radius 'p' is given in cartesian coordinates as:

$(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=p^{2}$

Here, it is given that the center of the sphere is at origin, i.e., at (0,0,0) & radius of the sphere is 5. That is, here we have,

$a=b=c=0,p=5$

That is, the equation of the sphere in cartesian coordinates is,

$(x-0)^{2}+(y-0)^{2}+(z-0)^{2}=5^{2}$

$\Rightarrow x^{2}+y^{2}+z^{2}=25$

Now, the cylindrical coordinate system is represented by $(r, \theta,z)$

The relation between cartesian and cylindrical coordinates is given by,

$x=rcos\theta,y=rsin\theta,z=z$

$r^{2}=x^{2}+y^{2},tan\theta=\frac{y}{x},z=z$

Thus, the obtained equation of the sphere in cartesian coordinates can be rewritten in cylindrical coordinates as,

$r^{2}+z^{2}=25$

This is the required equation of the given sphere in cylindrical coordinates.

(b) A cylinder is defined by the circle that gives the top and bottom faces or alternatively, the cross section, & it's axis. A cylinder running parallel to the z-axis has an axis that is parallel to the z-axis. The equation of such a cylinder is given by the equation of the circle of cross-section with the assumption that a point in 3 dimension lying on the cylinder has 'x' & 'y' values satisfying the equation of the circle & that 'z' can be any value.

That is, in cartesian coordinates, the equation of a cylinder running parallel to the z-axis having radius 'p' with center at (a, b) is given by,

$(x-a)^{2}+(y-b)^{2}=p^{2}$

Here, it is given that the center is at origin & radius is 1. That is, here, we have, $a=b=0,p=1$ . Then the equation of the cylinder in cartesian coordinates is,

$x^{2}+y^{2}=1$

Now, the spherical coordinate system is represented by $(\rho,\theta,\phi)$

The relation between cartesian and spherical coordinates is given by,

$x=\rho sin\phi cos\theta,y=\rho sin\phi sin\theta, z= \rho cos\phi$

Thus, the equation of the cylinder can be rewritten in spherical coordinates as,

$(\rho sin\phi cos\theta)^{2}+(\rho sin\phi sin\theta)^{2}=1$

$\Rightarrow \rho^{2} sin^{2}\phi cos^{2}\theta+\rho^{2} sin^{2}\phi sin^{2}\theta=1$

$\Rightarrow \rho^{2} sin^{2}\phi (cos^{2}\theta+sin^{2}\theta)=1$

$\Rightarrow \rho^{2} sin^{2}\phi=1$ (As $sin^{2}\theta+cos^{2}\theta=1$ )

Note that $\rho$ represents the distance of a point from the origin, which is always positive. $\phi$ represents the angle made by the line segment joining the point with z-axis. The range of $\phi$ is given as $0\leq \phi\leq \pi$ . We know that in this range the sine function is positive. Thus, we can say that $sin\phi$ is always positive.

Thus, we can square root both sides and only consider the positive root as,

$\Rightarrow \rho sin\phi=1$

This is the required equation of the cylinder in spherical coordinates.

Final answer:

(a) The equation of the given sphere in cylindrical coordinates is $r^{2}+z^{2}=25$

(b) The equation of the given cylinder in spherical coordinates is $\rho sin\phi=1$

7 0

3 years ago

Show how to solve this two step problem. My item is 40% off. The original price is X. Explain the two one step equations you wou

fredd [130]

40% of (times) X
4/10 of (times) X or 2/5 or .4
145 (times) .4 which is 58

4 0

3 years ago

Complete the equation of the graphed linear function in point-slope form.

vazorg [7]

The point-slope form:
$y-y_1=m(x-x_1)$
m - slope
x₁, y₁ - the coordinates of a point

It passes through the points (1,-2) and (2,2).
$(1,-2) \\
x_1=1 \\ y_1=-2 \\ \\
(2,2) \\
x_2=2 \\ y_2=2 \\ \\
m=\frac{y_2-y_1}{x_2-x_1}=\frac{2-(-2)}{2-1}=\frac{2+2}{1}=4$

$\boxed{y-(-2)=4(x-1)}$

3 0

3 years ago