Question

which solution method, graphing, substitution, or elimination, is the most appropriate for solving each system of equations? explain. 7. \\(\

$$\begin{cases} 3x + 8y = -4 \\\\ 2x - 4y = 16 \\end{cases}$$

\\) 8. \\(\

$$\begin{cases} 6x - y = 16 \\\\ x = (4y - 5) \\end{cases}$$

\\) 9. \\(\

$$\begin{cases} x + y = 19 \\\\ 3x - 2y = -3 \\end{cases}$$

\\)

Explanation:

System 7:

Step1: Identify method fit

Elimination is ideal here because we can easily scale the second equation to make the coefficients of $y$ opposites, allowing us to eliminate $y$ quickly.

Step2: Scale the second equation

Multiply $2x - 4y = 16$ by 2:
$2(2x - 4y) = 2(16) \implies 4x - 8y = 32$

Step3: Add to first equation

Add $3x + 8y = -4$ and $4x - 8y = 32$:
$(3x + 4x) + (8y - 8y) = -4 + 32 \implies 7x = 28$

Step4: Solve for $x$

$x = \frac{28}{7} = 4$

Step5: Substitute $x$ to find $y$

Plug $x=4$ into $2x - 4y = 16$:
$2(4) - 4y = 16 \implies 8 - 4y = 16 \implies -4y = 8 \implies y = -2$

System 8:

Step1: Identify method fit

Substitution is ideal here because one equation already gives $x$ directly in terms of $y$.

Step2: Substitute $x$ into first equation

Replace $x$ in $6x - y = 16$ with $4y - 5$:
$6(4y - 5) - y = 16$

Step3: Expand and simplify

$24y - 30 - y = 16 \implies 23y - 30 = 16$

Step4: Solve for $y$

$23y = 46 \implies y = \frac{46}{23} = 2$

Step5: Substitute $y$ to find $x$

Plug $y=2$ into $x=4y - 5$:
$x = 4(2) - 5 = 8 - 5 = 3$

System 9:

Step1: Identify method fit

Elimination is ideal here because we can scale the first equation to make the coefficients of $y$ opposites, or solve the first equation for $x$ for substitution. Elimination is efficient here.

Step2: Scale the first equation

Multiply $x + y = 19$ by 2:
$2(x + y) = 2(19) \implies 2x + 2y = 38$

Step3: Add to second equation

Add $2x + 2y = 38$ and $3x - 2y = -3$:
$(2x + 3x) + (2y - 2y) = 38 - 3 \implies 5x = 35$

Step4: Solve for $x$

$x = \frac{35}{5} = 7$

Step5: Substitute $x$ to find $y$

Plug $x=7$ into $x + y = 19$:
$7 + y = 19 \implies y = 19 - 7 = 12$

Answer:

1. System 7: Most appropriate method is Elimination. The system has coefficients that can be easily scaled to eliminate a variable, and the solution is $(4, -2)$.
2. System 8: Most appropriate method is Substitution. One equation already isolates $x$, so substituting directly into the other equation is efficient, and the solution is $(3, 2)$.
3. System 9: Most appropriate method is Elimination (or substitution, but elimination is efficient). We can scale the first equation to eliminate $y$, and the solution is $(7, 12)$.