To solve this problem, we’ll break it down into two main parts:
### Part 1: Graphing Line [tex]\( G \)[/tex]
Step 1: Find the Slope of Line [tex]\( G \)[/tex]
Line [tex]\( G \)[/tex] passes through the points [tex]\((-3, 5)\)[/tex] and [tex]\( (7, 1) \)[/tex]. The slope [tex]\( m \)[/tex] of a line is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given points:
[tex]\[ m_G = \frac{1 - 5}{7 + 3} = \frac{-4}{10} = -0.4 \][/tex]
Step 2: Determine the Equation of Line [tex]\( G \)[/tex]
The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex]. We already have [tex]\( m_G = -0.4 \)[/tex]. We need to find the y-intercept [tex]\( b \)[/tex]:
Using point [tex]\((-3, 5)\)[/tex]:
[tex]\[ 5 = -0.4(-3) + b \][/tex]
[tex]\[ 5 = 1.2 + b \][/tex]
[tex]\[ b = 3.8 \][/tex]
Thus, the equation of line [tex]\( G \)[/tex] is:
[tex]\[ y_G = -0.4x + 3.8 \][/tex]
### Part 2: Graphing Line [tex]\( H \)[/tex]
Line [tex]\( H \)[/tex] must satisfy the following conditions:
1. It intersects line [tex]\( G \)[/tex] at the point [tex]\((2, 3)\)[/tex].
2. Its slope [tex]\( m_H \)[/tex] is between 0 and 1.
3. Its y-intercept is an integer.
Step 1: Determine the Slope of Line [tex]\( H \)[/tex]
The slope of line [tex]\( H \)[/tex] should be greater than 0 and less than 1. Let's choose [tex]\( m_H = 0.5 \)[/tex], which meets the given condition.
Step 2: Determine the Y-Intercept of Line [tex]\( H \)[/tex]
Using the point of intersection [tex]\((2, 3)\)[/tex], and the chosen slope [tex]\( m_H = 0.5 \)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 3 = 0.5(2) + b \][/tex]
[tex]\[ 3 = 1 + b \][/tex]
[tex]\[ b = 2 \][/tex]
Equation of Line [tex]\( H \)[/tex]:
[tex]\[ y_H = 0.5x + 2 \][/tex]
### Summary
- The equation of line [tex]\( G \)[/tex] is [tex]\( y_G = -0.4x + 3.8 \)[/tex].
- The equation of line [tex]\( H \)[/tex] is [tex]\( y_H = 0.5x + 2 \)[/tex].
Both lines [tex]\( G \)[/tex] and [tex]\( H \)[/tex] intersect at the point [tex]\((2, 3)\)[/tex], and the slope of line [tex]\( H \)[/tex] is between 0 and 1 with a y-intercept that is an integer. You can now graph these lines on the coordinate plane accordingly.
