Question

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

$$\begin{cases}3x + 6y = -4 \\\\ 2x - 4y = 16\\end{cases}$$

\\) 8. \\(\

$$\begin{cases}8x - y = 18 \\\\ x = 4y - 5\\end{cases}$$

\\) 9. \\(\

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

\\) elimination substitution elimination

Explanation:

For System 7:

Neither variable is isolated, and we can easily scale the equations to eliminate one variable (e.g., multiply the second equation by 2 to make the coefficient of $y$ $-8$, which cancels with $8y$ in the first equation). This makes elimination efficient.

For System 8:

One equation directly gives $x$ defined in terms of $y$ ($x=4y-5$). We can substitute this expression for $x$ into the first equation, making substitution the simplest method.

For System 9:

While we could solve for $x$ or $y$ from the first equation for substitution, we can quickly eliminate $y$ by multiplying the first equation by 2, then adding it to the second equation. This avoids fractions that would come from substitution, making elimination more efficient.

Answer:

1. Elimination: No variables are isolated, and equations can be scaled easily to cancel a variable.
2. Substitution: One variable is already solved for, so substitution is direct.
3. Elimination: Avoids fractions from substitution, as equations can be scaled to cancel a variable simply.