Question

Given that 2g−e=3g(g−e), express g in terms of e in its simplest form.

Answer 1

To express g in terms of e, we need to solve the equation 2g−e=3g(g−e) and rearrange it. By factoring and dividing, we can simplify the expression for g in terms of e.

Step-by-step explanation:

To solve the equation 2g−e=3g(g−e) and express g in terms of e, we will distribute 3g to g and -e. This gives us 2g - e = 3g² - 3ge. Next, we will move all terms to one side of the equation to get 3g² - 2g - 3ge + e = 0. Factoring out g gives us g(3g - 2 - 3e) + e(1 - 3g) = 0. Finally, dividing both sides of the equation by (3g - 2 - 3e) gives us the simplified expression for g in terms of e: g = -e/(3g - 2 - 3e).

To express

�

g in terms of

�

e from the given equation

2

�

−

�

=

3

�

(

�

−

�

)

2g−e=3g(g−e), we'll start by simplifying the equation.

Distribute the

3

�

3g on the right side:

2

�

−

�

=

3

�

2

−

3

�

�

2g−e=3g

2

−3ge

Now, rearrange the terms to form a quadratic equation:

3

�

2

−

3

�

�

−

2

�

+

�

=

0

3g

2

−3ge−2g+e=0

Combine like terms:

3

�

2

−

(

3

�

+

2

)

�

+

�

=

0

3g

2

−(3e+2)g+e=0

Now, to solve for

�

g in terms of

�

e, you can use the quadratic formula:

�

=

−

�

±

�

2

−

4

�

�

2

�

g=

2a

−b±

b

2

−4ac

​

​

For this equation,

�

=

3

a=3,

�

=

−

(

3

�

+

2

)

b=−(3e+2), and

�

=

�

c=e. Substitute these values into the quadratic formula and simplify to obtain

�

g in terms of

�

e. The solution will be a quadratic expression in

�

e.

Answer 2

The solutions for g in terms of e are g= 2/3 or g=e.

Let's solve the equation step by step to express g in terms of e:

2g−e=3g(g−e)

First, distribute 3g across the parentheses on the right-hand side:

2g−e=3g2 −3ge

Rearrange this equation to set it equal to zero:

3g2 −3ge−2g+e=0

Now, let's factor out the g terms:

g(3g−2)−e(3g−2)=0

Now we have a common factor of (3g−2):

(3g−2)(g−e)=0

For this equation to hold true, either

(3g−2)=0 or (g−e)=0:

If 3g−2=0, then 3g=2, and solving for g gives g= 2/3

If g−e=0, then g=e.

So, the solutions for g in terms of e are g= 2/3 or g=e.