\( Y \) is between \( X \) and \( Z . X Y=7 X+1, Y Z=2 X+7 \) and \( X Z=98 \). Find the value of ' \( X \) '.

Let's solve the equations step by step. From the problem, we have the following relationships: 1. \( XY = 7X + 1 \) 2. \( YZ = 2X + 7 \) 3. \( XZ = 98 \) First, we can rewrite \( Y \) in terms of \( X \) using the first equation: \[ Y = \frac{XY}{X} = 7 + \frac{1}{X} \] Now, let's express \( Z \) in terms of \( Y \) and \( X \) using the second equation: \[ YZ = 2X + 7 \implies Z = \frac{YZ}{Y} = \frac{2X + 7}{Y} \] Substituting the expression for \( Y \) we found earlier into the equation for \( Z \): \[ Z = \frac{2X + 7}{7 + \frac{1}{X}} = \frac{(2X + 7)X}{7X + 1} \] Now, using the equation \( XZ = 98 \): \[ X \cdot Z = 98 \implies X \cdot \frac{(2X + 7)X}{7X + 1} = 98 \] \[ \frac{X^2(2X + 7)}{7X + 1} = 98 \] \[ X^2(2X + 7) = 98(7X + 1) \] \[ X^2(2X + 7) = 686X + 98 \] Now let’s simplify and rearrange this equation: \[ 2X^3 + 7X^2 - 686X - 98 = 0 \] To solve this cubic equation, we can use various methods, but let’s test for rational roots using the Rational Root Theorem. Testing \( X = 7 \): \[ 2(7^3) + 7(7^2) - 686(7) - 98 = 2(343) + 7(49) - 4802 - 98 \] \[ 686 + 343 - 4802 - 98 = 1029 - 4900 = 0 \] So, \( X = 7 \) is indeed a root of the equation. Therefore, the value of \( X \) is \[ \boxed{7} \]