Maya the trainer has two solo workout plans that she offers her clients: Plan A and Plan B. Each client does either one or the other (not both). On Monday there were $3$ clients who did Plan A and $5$ who did Plan B. On Tuesday there were $9$ clients who did Plan A and $7$ who did Plan B. Maya trained her Monday clients for a total of $6$ hours and her Tuesday clients for a total of $12$ hours. How long does each of the workout plans last?

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Q. Maya the trainer has two solo workout plans that she offers her clients: Plan A and Plan B. Each client does either one or the other (not both). On Monday there were

3

clients who did Plan A and

5

who did Plan B. On Tuesday there were

9

clients who did Plan A and

7

who did Plan B. Maya trained her Monday clients for a total of

6

hours and her Tuesday clients for a total of

12

hours. How long does each of the workout plans last?

Define Durations: Let's call the duration of Plan A $x$ hours and the duration of Plan B $y$ hours.
Monday Clients: On Monday, $3$ clients did Plan A and $5$ did Plan B, so the total time spent is $3x + 5y = 6$ hours.
Tuesday Clients: On Tuesday, $9$ clients did Plan A and $7$ did Plan B, so the total time spent is $9x + 7y = 12$ hours.
System of Equations: Now we have a system of equations: $\newline$ $1$ ) $3x + 5y = 6$ $\newline$ $2$ ) $9x + 7y = 12$
Eliminate x: Let's multiply the first equation by $3$ to make the x coefficients the same: $\newline$ $3(3x + 5y) = 3\cdot6$ $\newline$ $9x + 15y = 18$
Solve for y: Now we subtract the new equation from the second equation to eliminate x: $\newline$ $(9x + 7y) - (9x + 15y) = 12 - 18$ $\newline$ $-8y = -6$
Solve for x: Divide both sides by $-8$ to solve for y: $\newline$ y = $-6 / -8$ $\newline$ y = $3/4$
Clear Fraction: Now plug $y = \frac{3}{4}$ back into the first equation to solve for $x$ :
$3x + 5\left(\frac{3}{4}\right) = 6$
$3x + \frac{15}{4} = 6$
Subtract $15$ : Multiply everything by $4$ to clear the fraction: $\newline$ $4\times(3x) + 15 = 4\times6$ $\newline$ $12x + 15 = 24$
Subtract $15$ : Multiply everything by $4$ to clear the fraction: $\newline$ $4*(3x) + 15 = 4*6$ $\newline$ $12x + 15 = 24$ Subtract $15$ from both sides: $\newline$ $12x = 24 - 15$ $\newline$ $12x = 9$