Question

Enter the values for the highlighted variables to
complete the steps to find the sum:

Answer 1

The values for the variables are:

`\[ a = -1, \quad b = -9, \quad c = 9, \quad d = 3, \quad e = 3, \quad f = 2, \quad g = (3)/(2) \]`

Let's work through the steps to find the values of the variables
`\( a, b, c, d, e, f, g \)` in the given expression:

`\[ (3x)/(2x-6) + (9)/(6-2x) = (3x)/(2x-6) + (9)/(a(2x-6)) = (3x)/(2x-6) + (b)/(2x-6) = (3x-c)/(2x-6) = (d(x-e))/(f(x-3)) = g \]`

Step 1:

`\[ (3x)/(2x-6) + (9)/(6-2x) = (3x)/(2x-6) + (9)/(a(2x-6)) \]`

To make the denominators the same, set
`\( a = -1 \)`.

`\[ (3x)/(2x-6) + (9)/(6-2x) = (3x)/(2x-6) - (9)/(2x-6) \]`

Step 2:

`\[ (3x)/(2x-6) - (9)/(2x-6) = (3x)/(2x-6) + (b)/(2x-6) \]`

To make the numerators the same, set
`\( b = -9 \)`.

`\[ (3x)/(2x-6) - (9)/(2x-6) = (3x-9)/(2x-6) \]`

Step 3:

`\[ (3x-9)/(2x-6) = (3x-c)/(2x-6) \]`

To make the numerators the same, set
`\( c = 9 \)`.

`\[ (3x-9)/(2x-6) = (3x-9)/(2x-6) \]`

Step 4:

`\[ (3x-9)/(2x-6) = (d(x-e))/(f(x-3)) \]`

To make the denominators and numerators the same, set
`\( d = 3 \), \( e = 3 \), and \( f = 2 \)`.

`\[ (3x-9)/(2x-6) = (3(x-3))/(2(x-3)) \]`

Step 5:

`\[ (3(x-3))/(2(x-3)) = g \]`

To simplify the expression further, cancel out the common factor
`\( (x-3) \)`.

`\[ (3)/(2) = g \]`

So, the values for the variables are:

`\[ a = -1, \quad b = -9, \quad c = 9, \quad d = 3, \quad e = 3, \quad f = 2, \quad g = (3)/(2) \]`