Certainly! Let's solve each part of the question step-by-step.
### (i) Express [tex]$y$[/tex] in terms of [tex]$x$[/tex]
Since [tex]$y$[/tex] is directly proportional to [tex]$x$[/tex], we can write:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality. To find [tex]\( k \)[/tex], we use the given values [tex]\( y = 6 \)[/tex] when [tex]\( x = 2 \)[/tex]:
[tex]\[ 6 = k \cdot 2 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{6}{2} = 3 \][/tex]
Therefore, the expression for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[ y = 3x \][/tex]
### (ii) Find the value of [tex]$y$[/tex] when [tex]$x=11$[/tex]
Using the expression [tex]\( y = 3x \)[/tex], we substitute [tex]\( x = 11 \)[/tex]:
[tex]\[ y = 3 \cdot 11 = 33 \][/tex]
So, the value of [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex] is [tex]\( 33 \)[/tex].
### (iii) Calculate the value of [tex]$x$[/tex] when [tex]$y=12$[/tex]
Again, using the expression [tex]\( y = 3x \)[/tex], we substitute [tex]\( y = 12 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 12 = 3x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{12}{3} = 4 \][/tex]
So, the value of [tex]\( x \)[/tex] when [tex]\( y = 12 \)[/tex] is [tex]\( 4 \)[/tex].
### (iv) Draw the graph of [tex]$y$[/tex] against [tex]$x$[/tex]
The equation [tex]\( y = 3x \)[/tex] represents a straight line passing through the origin with a slope of 3. Here is what the graph should look like:
1. Draw the Cartesian plane with [tex]\( x \)[/tex] on the horizontal axis and [tex]\( y \)[/tex] on the vertical axis.
2. Plot the points we know:
- [tex]\( (2, 6) \)[/tex]
- [tex]\( (11, 33) \)[/tex]
- [tex]\( (4, 12) \)[/tex]
3. Draw the line passing through these points. The line should also pass through the origin [tex]\( (0, 0) \)[/tex].
#### Example Graph (No graph-drawing software available here, but the explanation covers the method):
To sketch this by hand:
- Plot the point [tex]\( (0, 0) \)[/tex].
- Plot point [tex]\( (2, 6) \)[/tex].
- Plot point [tex]\( (11, 33) \)[/tex].
- Plot point [tex]\( (4, 12) \)[/tex].
Connect these points with a straight line, which extends infinitely in both positive and negative directions along the line. The line should have a consistent slope, which shows that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 3 units, confirming our proportionality relationship [tex]\( y = 3x \)[/tex].
### Summary:
- The expression for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] is [tex]\( y = 3x \)[/tex].
- When [tex]\( x = 11 \)[/tex], [tex]\( y = 33 \)[/tex].
- When [tex]\( y = 12 \)[/tex], [tex]\( x = 4 \)[/tex].
- The graph is a straight line passing through the origin with a slope of 3.
This should give you full insight into solving and understanding the problem!
