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
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