Certainly! Let's tackle these problems one at a time.
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5.1 The gradient of the points [tex]\( P(5, a) \)[/tex] and [tex]\( Q(3, 4) \)[/tex] is [tex]\( -\frac{3}{2} \)[/tex]. Calculate [tex]\( a \)[/tex].
We know the gradient (or slope) formula for two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[ \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, the points are [tex]\( P(5, a) \)[/tex] and [tex]\( Q(3, 4) \)[/tex].
So,
- [tex]\((x_1, y_1) = (5, a)\)[/tex]
- [tex]\((x_2, y_2) = (3, 4)\)[/tex]
We are given that the gradient is [tex]\( -\frac{3}{2} \)[/tex].
Using the gradient formula:
[tex]\[ -\frac{3}{2} = \frac{4 - a}{3 - 5} \][/tex]
Simplify the denominator:
[tex]\[ -\frac{3}{2} = \frac{4 - a}{-2} \][/tex]
Multiply both sides by [tex]\(-2\)[/tex]:
[tex]\[ -\frac{3}{2} \times -2 = 4 - a \][/tex]
This simplifies to:
[tex]\[ 3 = 4 - a \][/tex]
Now, solve for [tex]\( a \)[/tex]:
[tex]\[ a = 4 - 3 \][/tex]
[tex]\[ a = 1 \][/tex]
So, the value of [tex]\( a \)[/tex] is [tex]\( 1 \)[/tex].
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5.2 The gradient of the points [tex]\( M(x, 5) \)[/tex] and [tex]\( N(-3, 4) \)[/tex] is [tex]\( -\frac{1}{4} \)[/tex]. Calculate [tex]\( x \)[/tex].
Again, using the gradient (slope) formula:
[tex]\[ \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, the points are [tex]\( M(x, 5) \)[/tex] and [tex]\( N(-3, 4) \)[/tex].
So,
- [tex]\((x_1, y_1) = (x, 5)\)[/tex]
- [tex]\((x_2, y_2) = (-3, 4)\)[/tex]
We are given that the gradient is [tex]\( -\frac{1}{4} \)[/tex].
Using the gradient formula:
[tex]\[ -\frac{1}{4} = \frac{4 - 5}{-3 - x} \][/tex]
Simplify the numerator:
[tex]\[ -\frac{1}{4} = \frac{-1}{-3 - x} \][/tex]
Since the numerator is [tex]\(-1\)[/tex], we can simplify:
[tex]\[ -\frac{1}{4} = \frac{-1}{-3 - x} \][/tex]
Which means:
[tex]\[ -\frac{1}{4} = -\frac{1}{3 + x} \][/tex]
We can now multiply both sides by [tex]\(-(3 + x)\)[/tex]:
[tex]\[ -\frac{1}{4}(3 + x) = -1 \][/tex]
This simplifies to:
[tex]\[ \frac{3 + x}{4} = 1 \][/tex]
Multiply both sides by 4:
[tex]\[ 3 + x = 4 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 4 - 3 \][/tex]
[tex]\[ x = -7 \][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( -7 \)[/tex].
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To summarize:
1. For part 5.1, the value of [tex]\( a \)[/tex] is [tex]\( 1 \)[/tex].
2. For part 5.2, the value of [tex]\( x \)[/tex] is [tex]\( -7 \)[/tex].
