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Answer :
To find the constant of proportionality [tex]\( r \)[/tex] in the equation [tex]\( y = rx \)[/tex], we need to look at the relationship between the quantities [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Here are the given pairs of values:
- [tex]\( x = 5.8 \)[/tex], [tex]\( y = 5.8 \)[/tex]
- [tex]\( x = 7.5 \)[/tex], [tex]\( y = 7.5 \)[/tex]
- [tex]\( x = 11.2 \)[/tex], [tex]\( y = 11.2 \)[/tex]
Since [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are proportional, this means that [tex]\( y = rx \)[/tex].
To find the constant [tex]\( r \)[/tex], we use one pair of values to solve for [tex]\( r \)[/tex]. Let's use the first pair:
[tex]\[
r = \frac{y}{x} = \frac{5.8}{5.8}
\][/tex]
Calculate [tex]\( r \)[/tex]:
[tex]\[
r = 1.0
\][/tex]
Now, let's check that this value holds for the other pairs:
For the second pair:
[tex]\[
r = \frac{y}{x} = \frac{7.5}{7.5} = 1.0
\][/tex]
For the third pair:
[tex]\[
r = \frac{y}{x} = \frac{11.2}{11.2} = 1.0
\][/tex]
In all cases, the constant of proportionality [tex]\( r \)[/tex] is the same. Thus, the constant of proportionality [tex]\( r \)[/tex] is [tex]\( 1.0 \)[/tex].
This means the equation [tex]\( y = rx \)[/tex] simplifies to [tex]\( y = 1.0 \times x \)[/tex], or simply [tex]\( y = x \)[/tex].
Here are the given pairs of values:
- [tex]\( x = 5.8 \)[/tex], [tex]\( y = 5.8 \)[/tex]
- [tex]\( x = 7.5 \)[/tex], [tex]\( y = 7.5 \)[/tex]
- [tex]\( x = 11.2 \)[/tex], [tex]\( y = 11.2 \)[/tex]
Since [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are proportional, this means that [tex]\( y = rx \)[/tex].
To find the constant [tex]\( r \)[/tex], we use one pair of values to solve for [tex]\( r \)[/tex]. Let's use the first pair:
[tex]\[
r = \frac{y}{x} = \frac{5.8}{5.8}
\][/tex]
Calculate [tex]\( r \)[/tex]:
[tex]\[
r = 1.0
\][/tex]
Now, let's check that this value holds for the other pairs:
For the second pair:
[tex]\[
r = \frac{y}{x} = \frac{7.5}{7.5} = 1.0
\][/tex]
For the third pair:
[tex]\[
r = \frac{y}{x} = \frac{11.2}{11.2} = 1.0
\][/tex]
In all cases, the constant of proportionality [tex]\( r \)[/tex] is the same. Thus, the constant of proportionality [tex]\( r \)[/tex] is [tex]\( 1.0 \)[/tex].
This means the equation [tex]\( y = rx \)[/tex] simplifies to [tex]\( y = 1.0 \times x \)[/tex], or simply [tex]\( y = x \)[/tex].
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