Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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!
### (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!
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
