To determine the value of [tex]\( c \)[/tex] where the gradient (or slope) between the points [tex]\( X(-1, -1) \)[/tex] and [tex]\( Y(c, -2) \)[/tex] is [tex]\(\frac{5}{2}\)[/tex], we need to use the formula for the gradient between two points, [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
\text{gradient} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\( (x_1, y_1) = (-1, -1) \)[/tex] and [tex]\( (x_2, y_2) = (c, -2) \)[/tex]. Plugging these coordinates into the gradient formula, we get:
[tex]\[
\frac{-2 - (-1)}{c - (-1)} = \frac{5}{2}
\][/tex]
This simplifies to:
[tex]\[
\frac{-2 + 1}{c + 1} = \frac{5}{2}
\][/tex]
Which reduces to:
[tex]\[
\frac{-1}{c + 1} = \frac{5}{2}
\][/tex]
To find [tex]\( c \)[/tex], we will cross-multiply to get rid of the fractions:
[tex]\[
-1 \cdot 2 = 5 \cdot (c + 1)
\][/tex]
So,
[tex]\[
-2 = 5c + 5
\][/tex]
Next, isolate [tex]\( c \)[/tex] by moving all terms involving [tex]\( c \)[/tex] to one side of the equation and constant terms to the other side:
[tex]\[
-2 - 5 = 5c
\][/tex]
[tex]\[
-7 = 5c
\][/tex]
Finally, divide both sides by 5 to solve for [tex]\( c \)[/tex]:
[tex]\[
c = \frac{-7}{5} = -1.4
\][/tex]
Thus, the value of [tex]\( c \)[/tex] is [tex]\(-1.4\)[/tex].
