Solve for x:

(x - 1)/(x - t - 1) + (x - 6 t - 1)/(x - 7 t - 1) = 22/7

Write the left hand side as a single fraction.

Bring (x - 6 t - 1)/(x - 7 t - 1) + (x - 1)/(x - t - 1) together using the common denominator (-x + t + 1) (-x + 7 t + 1):

(2 (x^2 - 7 t x - 2 x + 3 t^2 + 7 t + 1))/((-x + t + 1) (-x + 7 t + 1)) = 22/7

Multiply both sides by a polynomial with respect to x to clear fractions.

Cross multiply:

14 (x^2 - 7 t x - 2 x + 3 t^2 + 7 t + 1) = 22 (-x + t + 1) (-x + 7 t + 1)

Write the quadratic polynomial on the left hand side in standard form.

Expand and collect in terms of x:

14 + 98 t + 42 t^2 + x (-98 t - 28) + 14 x^2 = 22 (-x + t + 1) (-x + 7 t + 1)

Write the quadratic polynomial on the right hand side in standard form.

Expand and collect in terms of x:

14 + 98 t + 42 t^2 + x (-98 t - 28) + 14 x^2 = 22 + 176 t + 154 t^2 + x (-176 t - 44) + 22 x^2

Move everything to the left hand side.

Subtract 22 + 176 t + 154 t^2 + (-176 t - 44) x + 22 x^2 from both sides:

-8 - 78 t - 112 t^2 - x (-176 t - 44) + x (-98 t - 28) - 8 x^2 = 0

Write the quadratic polynomial on the left hand side in standard form.

Expand and collect in terms of x:

-8 - 78 t - 112 t^2 + x (78 t + 16) - 8 x^2 = 0

Factor the left hand side.

The left hand side factors into a product with three terms:

-2 (-4 x + 7 t + 4) (-x + 8 t + 1) = 0

Divide both sides by a constant to simplify the equation.

Divide both sides by -2:

(-4 x + 7 t + 4) (-x + 8 t + 1) = 0

Find the roots of each term in the product separately.

Split into two equations:

-4 x + 7 t + 4 = 0 or -x + 8 t + 1 = 0

Look at the first equation: Isolate terms with x to the left hand side.

Subtract 7 t + 4 from both sides:

-4 x = -7 t - 4 or -x + 8 t + 1 = 0

Solve for x.

Divide both sides by -4:

x = (7 t)/4 + 1 or -x + 8 t + 1 = 0

Look at the second equation: Isolate terms with x to the left hand side.

Subtract 8 t + 1 from both sides:

x = (7 t)/4 + 1 or -x = -8 t - 1

Solve for x.

Multiply both sides by -1:

Answer:

x = (7 t)/4 + 1 or x = 8 t + 1