105(974). Determine the value of $b$, if the graph of the function defined by the formula passes through the point with coordinates $(3 ; 10)$:

1) $y=x+b$;
2) $y=3 x+b$
3) $y=-\frac{1}{3} x+b$;
4) $y=-\frac{1}{2} x+b$.

A n s w e r: 1) $b=7$; 2) $b=1$; 3) $b=11$; 4) $b=11.5$.

106(975). Define the function by a formula whose graph is a straight line passing through points $A$ and $B$:

1) $A(-6 ;-3), B(2 ;-3)$
2) $A(-4 ;-4), B(3,3)$
3) $A(2 ; 2), B(0 ; 4)$;
4) $A(3 ;-8), B(-5 ; 32)$.

A n s w e r: 1) $y=-3$; 2) $y=x$; 3) $y=-x+4$; 4) $y=-5 x+7$.

N o t e. During club sessions, students can be introduced to the equation of a straight line passing through two points with given coordinates. Otherwise, solving such types of problems reduces to solving a system of two linear equations with two unknowns, which we obtain by substituting the coordinates of the given points into the equation of the line $y=k x+b$. Thus, solving problem (4) we get the system:

$$
\left\{\begin{array}{l}
-8=3 k+b \\
32=-5 k+b
\end{array}\right.
$$

solving which we find $k=-5, b=7$, hence $y=-5 x+7$.

$\triangle$ 107(976). A cylindrical helical spring of height 122 mm is to be made from steel wire with a diameter of 5 mm. Find the number of turns of the spring, if the gap between the turns of the unloaded spring should be 8 mm.
Solution. Let $x$ be the number of turns in the spring. Considering that the number of gaps in the spring is one less than the number of turns, according to the problem, we set up the equation $5x + 8(x-1) = 122$, from which $x = 10$.

Answer: 10 turns.