Question

1. a system of equations is graphed on the grid.

which system of equations are best represented by the graph?
select two correct answers.
$y = 3x - 2$
$y = -x + 2$
$y = x + 2$
$y = 3(x + 2)$
$y = 3x + 5$

Explanation:

Step1: Analyze the first line (steeper line)

The steeper line has a positive slope and a y - intercept. From the graph, when \(x = 0\), \(y=5\) (y - intercept \(b = 5\)) and the slope \(m\) can be calculated. Let's take two points, say \((0,5)\) and \((- 1,2)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 5}{-1-0}=\frac{-3}{-1} = 3\). So the equation of this line is \(y=3x + 5\) (using \(y=mx + b\) with \(m = 3\) and \(b = 5\)) or we can also expand \(y = 3(x + 2)-1=3x+6 - 1=3x + 5\), but \(y = 3x+5\) is a direct form. Also, \(y=3(x + 2)\) expands to \(y=3x + 6\) which does not match, \(y = 3x-2\) has \(b=-2\) which is wrong.

Step2: Analyze the second line (less steep line)

The second line has a negative slope. When \(x = 0\), \(y = 2\) (y - intercept \(b = 2\)) and let's take two points \((0,2)\) and \((2,0)\). The slope \(m=\frac{0 - 2}{2-0}=\frac{-2}{2}=-1\). So the equation of this line is \(y=-x + 2\) (using \(y=mx + b\) with \(m=-1\) and \(b = 2\)). \(y=x + 2\) has a positive slope, so it's wrong.

Answer:

B. \(y=-x + 2\), E. \(y = 3x+5\) (assuming the options are labeled as A: \(y = 3x-2\), B: \(y=-x + 2\), C: \(y=x + 2\), D: \(y = 3(x + 2)\), E: \(y = 3x+5\))