1. **State the problem:** Solve each system of equations using substitution. Identify which three systems have the solution $(4,16)$ and unscramble their letters to find a secret code.
2. **Recall substitution method:** Solve one equation for one variable, then substitute into the other equation.
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### System T:
\begin{cases}
7x + y = 12 \\
y = 4x - 21
\end{cases}
Substitute $y$ from second into first:
$$7x + (4x - 21) = 12$$
$$11x - 21 = 12$$
$$11x = 33$$
$$x = 3$$
Then $y = 4(3) - 21 = 12 - 21 = -9$
Solution: $(3, -9)$
### System E:
\begin{cases}
2x + 3y = 18 \\
y = -6x - 2
\end{cases}
Substitute $y$:
$$2x + 3(-6x - 2) = 18$$
$$2x - 18x - 6 = 18$$
$$-16x = 24$$
$$x = -\frac{24}{16} = -\frac{3}{2}$$
Then $y = -6(-\frac{3}{2}) - 2 = 9 - 2 = 7$
Solution: $(-\frac{3}{2}, 7)$
### System U:
\begin{cases}
-3x + \frac{1}{4}y = -8 \\
8x - y = 16
\end{cases}
From second: $y = 8x - 16$
Substitute into first:
$$-3x + \frac{1}{4}(8x - 16) = -8$$
$$-3x + 2x - 4 = -8$$
$$-x - 4 = -8$$
$$-x = -4$$
$$x = 4$$
Then $y = 8(4) - 16 = 32 - 16 = 16$
Solution: $(4,16)$
### System R:
\begin{cases}
11x + 9y = -20 \\
x = -5y - 6
\end{cases}
Substitute $x$:
$$11(-5y - 6) + 9y = -20$$
$$-55y - 66 + 9y = -20$$
$$-46y - 66 = -20$$
$$-46y = 46$$
$$y = -1$$
Then $x = -5(-1) - 6 = 5 - 6 = -1$
Solution: $(-1, -1)$
### System B:
\begin{cases}
-2x + y = 8 \\
\frac{1}{2}x + 3y = 50
\end{cases}
From first: $y = 2x + 8$
Substitute into second:
$$\frac{1}{2}x + 3(2x + 8) = 50$$
$$\frac{1}{2}x + 6x + 24 = 50$$
$$6.5x = 26$$
$$x = 4$$
Then $y = 2(4) + 8 = 8 + 8 = 16$
Solution: $(4,16)$
### System P:
\begin{cases}
5x - 3y = -4 \\
y = \frac{5}{3}x + 6
\end{cases}
Substitute $y$:
$$5x - 3\left(\frac{5}{3}x + 6\right) = -4$$
$$5x - 5x - 18 = -4$$
$$-18 = -4$$
No solution (contradiction)
---
### Word Problem S (Piggy bank):
\begin{cases}
x + y = 20 \\
0.10x + 0.25y = 4.40
\end{cases}
From first: $y = 20 - x$
Substitute:
$$0.10x + 0.25(20 - x) = 4.40$$
$$0.10x + 5 - 0.25x = 4.40$$
$$-0.15x = -0.60$$
$$x = 4$$
Then $y = 20 - 4 = 16$
Number of dimes: 4, quarters: 16
### Word Problem A (Package):
\begin{cases}
x + y = 14 \\
x + 0.5y = 9
\end{cases}
From first: $y = 14 - x$
Substitute:
$$x + 0.5(14 - x) = 9$$
$$x + 7 - 0.5x = 9$$
$$0.5x = 2$$
$$x = 4$$
Then $y = 14 - 4 = 10$
Number of magazines: 4, chips: 10
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### Secret code:
Systems with solution $(4,16)$ are U, B, and S.
Unscramble letters: **SUB**
**Final answers:**
- Solutions for each system as above.
- Secret code: **SUB**
Systems Substitution
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