Absolutely, let's go through the problem step-by-step.
(i) Express the above statements in the form of linear equations:
Let's define:
- [tex]\( b \)[/tex] as the cost of the bag.
- [tex]\( u \)[/tex] as the cost of the umbrella.
According to the problem, we have two pieces of information:
1. The total cost of the bag and the umbrella is Rs 1000:
[tex]\[
b + u = 1000
\][/tex]
2. The umbrella is cheaper than the bag by Rs 200:
[tex]\[
b = u + 200
\][/tex]
So, the system of linear equations is:
[tex]\[
\begin{cases}
b + u = 1000 \\
b = u + 200
\end{cases}
\][/tex]
(ii) Find the cost of the bag and the umbrella:
First, we'll substitute the second equation [tex]\( b = u + 200 \)[/tex] into the first equation [tex]\( b + u = 1000 \)[/tex]:
Substitute [tex]\( b \)[/tex] in [tex]\( b + u = 1000 \)[/tex]:
[tex]\[
(u + 200) + u = 1000
\][/tex]
Simplify the equation:
[tex]\[
u + 200 + u = 1000
\][/tex]
[tex]\[
2u + 200 = 1000
\][/tex]
Next, we'll solve for [tex]\( u \)[/tex] by isolating [tex]\( u \)[/tex]:
[tex]\[
2u = 1000 - 200
\][/tex]
[tex]\[
2u = 800
\][/tex]
[tex]\[
u = 400
\][/tex]
So, the cost of the umbrella [tex]\( u \)[/tex] is Rs 400.
To find the cost of the bag [tex]\( b \)[/tex], we use the equation [tex]\( b = u + 200 \)[/tex]:
[tex]\[
b = 400 + 200
\][/tex]
[tex]\[
b = 600
\][/tex]
Therefore, the cost of the bag [tex]\( b \)[/tex] is Rs 600 and the cost of the umbrella [tex]\( u \)[/tex] is Rs 400.
Conclusion:
- The cost of the bag is Rs 600.
- The cost of the umbrella is Rs 400.
