Question

each side of a triangle lies along a line in a coordinate plane. the three lines that contain the sides are represented by the equations given to the right. complete parts a to c below. equation 1: $x - 2y = -4$; equation 2: $2x + y = -3$; equation 3: $7x - 4y = 12$. then there are multiple-choice options with systems of equations: a. $\

$$\begin{cases}2x + y = -4 \\\\ 7x - 4y = 12\\end{cases}$$

$; b. $\

$$\begin{cases}x - 2y = -4 \\\\ 7x - 4y = 12\\end{cases}$$

$; c. $\

$$\begin{cases}7x - 4y = 12 \\\\ 2x + y = -3\\end{cases}$$

$; d. $\

$$\begin{cases}x - 2y = -4 \\\\ 7x - 4y = -3\\end{cases}$$

$; e. $\

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

$; f. $\

$$\begin{cases}2x + y = 12 \\\\ 7x - 4y = -3\\end{cases}$$

$; g. (partially visible system); h. $\

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

$

Explanation:

Step1: Identify Given Equations

The three lines (sides of the triangle) have equations:
Equation 1: \( x - 2y = -4 \)
Equation 2: \( 2x + y = -3 \)
Equation 3: \( 7x - 4y = 12 \)

Step2: Analyze Each Option

• Option A: \(

$$\begin{cases} 2x + y = -4 \\ 7x - 4y = 12 \end{cases}$$

\) → Equation 2 is incorrect (should be \( -3 \), not \( -4 \)). Eliminate.

• Option B: \(

$$\begin{cases} x - 2y = -4 \\ 7x - 4y = 12 \end{cases}$$

\) → Missing Equation 2 (\( 2x + y = -3 \)). Eliminate.

• Option C: \(

$$\begin{cases} 7x - 4y = 12 \\ 2x + y = -3 \end{cases}$$

\) → Missing Equation 1 (\( x - 2y = -4 \)). Eliminate.

• Option E: \(

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

\) → Contains Equation 1 and Equation 2 (two of the three lines). Wait, but let’s check others. Wait, no—wait, the problem is about the three lines, but maybe the options are pairs? Wait, no, the original three equations are Equation 1: \( x - 2y = -4 \), Equation 2: \( 2x + y = -3 \), Equation 3: \( 7x - 4y = 12 \). Wait, maybe the options are systems of two equations (intersections of two sides, i.e., vertices). Wait, no, the question is likely about which system uses two of the three given equations. Wait, Option E has \( x - 2y = -4 \) (Equation 1) and \( 2x + y = -3 \) (Equation 2) → these are two of the three lines. Let’s check other options:

• Option D: Incorrect equations (e.g., \( x - 2y = -3 \) is not given). Eliminate.
• Option Q: \( 7x - 4y = -3 \) is incorrect (given is \( 12 \)). Eliminate.
• Option F: \( 2x + y = 12 \) and \( 7x - 4y = -3 \) are incorrect. Eliminate.
• Option H: \( x - 2y = -3 \) and \( 2x + y = -4 \) are incorrect. Eliminate.

Wait, correction: The three lines are Equation 1 (\( x - 2y = -4 \)), Equation 2 (\( 2x + y = -3 \)), Equation 3 (\( 7x - 4y = 12 \)). So a valid system should use two of these three. Option E uses Equation 1 and Equation 2 (both given). Let’s confirm:

• Equation 1: \( x - 2y = -4 \) (matches)
• Equation 2: \( 2x + y = -3 \) (matches)

Other options have incorrect equations. So Option E is correct.

Answer:

E. \(

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

\)