Let's break down and solve each part of the problem step by step.
### Question 5.1
Given:
- Point [tex]\( P(5, a) \)[/tex]
- Point [tex]\( Q(3, 4) \)[/tex]
- Gradient (slope) of line [tex]\( PQ = -\frac{3}{2} \)[/tex]
We use the formula for the gradient (slope) of a line passing through 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]
Substitute the given values:
[tex]\[
-\frac{3}{2} = \frac{4 - a}{3 - 5}
\][/tex]
Simplify the denominator:
[tex]\[
-\frac{3}{2} = \frac{4 - a}{-2}
\][/tex]
To eliminate the fraction, multiply both sides by [tex]\(-2\)[/tex]:
[tex]\[
-2 \times -\frac{3}{2} = 4 - a
\][/tex]
This simplifies to:
[tex]\[
3 = 4 - a
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 4 - 3
\][/tex]
[tex]\[
a = 1
\][/tex]
Thus, the value of [tex]\( a \)[/tex] is:
[tex]\[
a = 1
\][/tex]
### Question 5.2
Given:
- Point [tex]\( M(x, 5) \)[/tex]
- Point [tex]\( N(-3, 4) \)[/tex]
- Gradient (slope) of line [tex]\( MN = -\frac{1}{4} \)[/tex]
We use the same gradient formula for the line passing through points [tex]\( M \)[/tex] and [tex]\( N \)[/tex]:
[tex]\[
\text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the given values:
[tex]\[
-\frac{1}{4} = \frac{4 - 5}{-3 - x}
\][/tex]
Simplify the numerator:
[tex]\[
-\frac{1}{4} = \frac{-1}{-3 - x}
\][/tex]
To eliminate the fraction, we can set up the equation and cross-multiply:
[tex]\[
-1 \times 4 = -1 \times (-3 - x)
\][/tex]
Simplify both sides:
[tex]\[
-4 = 3 + x
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -4 - 3
\][/tex]
[tex]\[
x = -7
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = -7
\][/tex]
### Summary
- 5.1 The value of [tex]\( a \)[/tex] is [tex]\( 1 \)[/tex].
- 5.2 The value of [tex]\( x \)[/tex] is [tex]\( -7 \)[/tex].
