Sure, let's proceed step-by-step to find how many packs of medium and large cups Ivy bought:
1. Define Variables:
- Let [tex]\( M \)[/tex] represent the number of packs of medium cups.
- Let [tex]\( L \)[/tex] represent the number of packs of large cups.
2. Formulate Equations Based on the Problem:
- The first equation comes from the total number of packs Ivy bought:
[tex]\[
M + L = 10
\][/tex]
- The second equation comes from the total cost of the packs. Each medium pack costs \[tex]$2 and each large pack costs \$[/tex]3, and Ivy spent \$26 in total:
[tex]\[
2M + 3L = 26
\][/tex]
3. Solving the System of Equations:
- Equation 1: [tex]\( M + L = 10 \)[/tex]
- Equation 2: [tex]\( 2M + 3L = 26 \)[/tex]
4. Substitute [tex]\( L \)[/tex] from Equation 1 into Equation 2:
- From Equation 1, solve for [tex]\( L \)[/tex]:
[tex]\[
L = 10 - M
\][/tex]
- Substitute [tex]\( L = 10 - M \)[/tex] in Equation 2:
[tex]\[
2M + 3(10 - M) = 26
\][/tex]
- Simplify and solve for [tex]\( M \)[/tex]:
[tex]\[
2M + 30 - 3M = 26 \\
-M + 30 = 26 \\
-M = 26 - 30 \\
-M = -4 \\
M = 4
\][/tex]
5. Find the value of [tex]\( L \)[/tex]:
- Substitute [tex]\( M = 4 \)[/tex] back into [tex]\( L = 10 - M \)[/tex]:
[tex]\[
L = 10 - 4 \\
L = 6
\][/tex]
Hence, Ivy bought 4 packs of medium cups and 6 packs of large cups.
The final equations and solutions are:
[tex]\(
M + L = 10 \quad \text { (where } M = 4 \text { and } L = 6)
\)[/tex]
