To find the value of [tex]\( b \)[/tex] for which the gradient (or slope) between the points [tex]\( C(-6, 2) \)[/tex] and [tex]\( D(b, -8) \)[/tex] is -5, follow these steps:
1. Write down the formula for the gradient:
The gradient [tex]\( m \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Substitute the known coordinates and the gradient:
Here, the coordinates are [tex]\( C(-6, 2) \)[/tex] and [tex]\( D(b, -8) \)[/tex], and the gradient [tex]\( m \)[/tex] is -5. Therefore:
[tex]\[
-5 = \frac{-8 - 2}{b - (-6)}
\][/tex]
3. Simplify the equation:
Simplify the right-hand side of the equation:
[tex]\[
-5 = \frac{-10}{b + 6}
\][/tex]
4. Solve for [tex]\( b \)[/tex]:
To isolate [tex]\( b \)[/tex], cross-multiply to remove the fraction:
[tex]\[
-5(b + 6) = -10
\][/tex]
Distribute the -5:
[tex]\[
-5b - 30 = -10
\][/tex]
Add 30 to both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
-5b = 20
\][/tex]
Divide both sides by -5:
[tex]\[
b = -4
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] that satisfies the condition is:
[tex]\[
b = -4
\][/tex]
