Question

1. the tables of ordered pairs represent some points on the graphs of two different lines.

line a

x	-11	-6	-1	4
y	81	51	21	-9

line b

x	-9	-4	1	6
y	18	3	-12	-27

which system of equations represents the graph of these two lines?
options:

• $x - 6y = -90$; $x - 3y = 27$
• $x + 6y = 90$; $x + 3y = -27$
• $6x - y = -15$; $3x - y = 9$
• $6x + y = 15$; $3x + y = -9$

clear all

Explanation:

Step1: Find slope of Line a

Use two points from Line a, e.g., \((-11, 81)\) and \((-6, 51)\).
Slope \(m_a = \frac{51 - 81}{-6 - (-11)} = \frac{-30}{5} = -6\).
Equation form: \(y - 81 = -6(x + 11)\)
Simplify: \(y = -6x - 66 + 81 \implies y = -6x + 15\).
Rearrange: \(6x + y = 15\).

Step2: Find slope of Line b

Use two points from Line b, e.g., \((-9, 18)\) and \((-4, 3)\).
Slope \(m_b = \frac{3 - 18}{-4 - (-9)} = \frac{-15}{5} = -3\).
Equation form: \(y - 18 = -3(x + 9)\)
Simplify: \(y = -3x - 27 + 18 \implies y = -3x - 9\).
Rearrange: \(3x + y = -9\).

Step3: Match with options

The equations \(6x + y = 15\) (Line a) and \(3x + y = -9\) (Line b) match the last option.

Answer:

\(6x + y = 15\)
\(3x + y = -9\) (the last option with these two equations)