## Algebra 2 Common Core

$7$ One dollar bills, $8$ Five dollar bills.
Let $n =$ number of one dollar bills, and $f=$ number of five dollar bills. Knowing there are a total of 15 bills, then $n + f = 15$ (Equation 1) Knowing there is a total of 47 dollars, then the total value of the bills is: $(1)n + 5f = 47 \\n +5f = 47$ (Equation 2) Solving by elimination: Subtract Equation 1 to Equation 2 to cancel out the $n$'s giving us: $n + 5f - (n + f) = 47 -15 \\n+5f-n-f=32 \\4f=32$ Divide both sides of the equation by 4: $\frac{4f}{4} = \frac{32}{4} \\f=8$ Substitute $f = 8$ to $n + f = 15$ and solve: $n+f=15 \\n + 8 = 15 \\n = 15-8 \\n=7$ Thus, there are $7$ one dollar bills and $8$ five dollar bills. Alternatively, solving by substitution: Make $f$ the subject by subtracting $n$ to both sides of the Equation 1, giving $f = 15 - n$ Substitute $15 - n$ to $f$ in $n + 5f = 47$: $n+5f=47 \\n + 5(15 - n) = 47$ Distribute 5: $n + 75 - 5n = 47$ Collect like terms: $-4n + 75 = 47$ Subtract $75$ to both sides: $-4n = -28$ Divide both sides by $-4$: $n = 7$ Substitute in $n = 7$ to $f = 15 - n$ giving $f = 15 - 7 = 8$ Thus, $n = 7$, $f = 8$