\( \left. \begin{array} { l l } { a = ? } & { T 7 = a + ( 7 - 1 ) d } \\ { d = 7 } & { T 7 = a + 6 d = - 20 } \\ { d = ? } & { } \\ { } & { 120 = a t } \end{array} \right. \)

Let's break this down step by step in an easy and fun way! First, we know from the equation \(T_7 = a + (7 - 1)d\) or simplified to \(T_7 = a + 6d\). Also, we're given that \(d = 7\). Plugging that into the equation gives us: \[T_7 = a + 6(7) = a + 42\] We know \(T_7 = -20\), so we can set up the equation: \[ a + 42 = -20 \] Solving for \(a\) gives us: \[ a = -20 - 42 = -62 \] Now that we have \(a\), we can use it to find the relationship with \(t\) as per the equation \(120 = a t\). Plugging \(a\) into the equation: \[ 120 = -62 t \] To find \(t\), we rearrange it to: \[ t = \frac{120}{-62} \approx -1.935 \] And there you have it! The values are \(a = -62\) and \(t \approx -1.935\). Mathematics can be like a treasure hunt—solving these equations is just uncovering the hidden gems!