Alright, let's break down the problem step-by-step using the information provided.
1. Form the equations based on the given information:
Let:
- [tex]\( s \)[/tex] be the cost of one shirt in Ksh.
- [tex]\( t \)[/tex] be the cost of one pair of trousers in Ksh.
First scenario:
- Danny buys 2 shirts and 3 pairs of trousers for a total of Ksh. 10,500.
This can be expressed mathematically as:
[tex]\[ 2s + 3t = 10500 \tag{1} \][/tex]
Second scenario:
- Danny buys 3 shirts and 2 pairs of trousers, and he is left with Ksh. 1000.
Since he had originally Ksh. 10,500, the amount he spends is:
[tex]\[ 10500 - 1000 = 9500 \text{ Ksh} \][/tex]
This can be expressed mathematically as:
[tex]\[ 3s + 2t = 9500 \tag{2} \][/tex]
So we have the following system of equations:
[tex]\[
\begin{cases}
2s + 3t = 10500 \tag{1} \\
3s + 2t = 9500 \tag{2}
\end{cases}
\][/tex]
2. Solve the system of simultaneous equations using the elimination method:
First, we multiply equation (1) by 3 and equation (2) by 2, in order to eliminate one of the variables when we subtract the equations:
[tex]\[
\begin{cases}
3(2s + 3t) = 3(10500) \implies 6s + 9t = 31500 \tag{3} \\
2(3s + 2t) = 2(9500) \implies 6s + 4t = 19000 \tag{4}
\end{cases}
\][/tex]
Now, subtract equation (4) from equation (3):
[tex]\[
(6s + 9t) - (6s + 4t) = 31500 - 19000
\][/tex]
Simplify:
[tex]\[
5t = 12500
\][/tex]
Divide both sides by 5:
[tex]\[
t = 2500
\][/tex]
So the cost of one pair of trousers is Ksh. 2500.
3. Find the cost of one shirt by substituting [tex]\( t = 2500 \)[/tex] into one of the original equations:
Substitute [tex]\( t = 2500 \)[/tex] into equation (1):
[tex]\[
2s + 3(2500) = 10500
\][/tex]
Simplify:
[tex]\[
2s + 7500 = 10500
\][/tex]
Subtract 7500 from both sides:
[tex]\[
2s = 3000
\][/tex]
Divide by 2:
[tex]\[
s = 1500
\][/tex]
So the cost of one shirt is Ksh. 1500.
4. Verify the solution by substituting both [tex]\( s \)[/tex] and [tex]\( t \)[/tex] back into the second equation:
Using the values [tex]\( s = 1500 \)[/tex] and [tex]\( t = 2500 \)[/tex] in equation (2):
[tex]\[
3(1500) + 2(2500) = 4500 + 5000 = 9500
\][/tex]
The solution satisfies both equations, confirming our values.
Answer:
- The cost of one shirt is Ksh. 1500.
- The cost of one pair of trousers is Ksh. 2500.
