Answer:
A, B, C, E, F
Step-by-step explanation:
In order to identify the applicable statements to the given graph, we must first determine the slope and y-intercept of the equation.
Slope
Choose two points from the graph.
Let (x₁, y₁) = (-1, 1)
(x₂, y₂) = (3, 2)
Substitute these coordinates into the following slope formula:
m = (y₂ - y₁)/(x₂ - x₁)
[tex]m = \frac{2 - 1}{3 - (-1)} = \frac{1}{3 + 1} = \frac{1}{4}[/tex]
Therefore, the slope of the line is, m = ¼.
Y-intercept
Next, we need to determine the y-intercept of the equation. The y-intercept is the y-coordinate of the point (0, b ) where the graph crosses the y-axis. The y-intercept also provides the value of y when x = 0. There are two ways of identifying the y-intercept: by referencing the graph, or solving for the value of b algebraically.
To solve for the y-intercept algebraically, use the slope, m = ¼, and one of the points on the graph, (-1, 1). Substitute these values into the following slope-intercept form: y = mx + b.
y = mx + b
1 = ¼(-1) + b
1 = -¼ + b
Add ¼ to both sides of the equation:
1 + ¼ = -¼ + ¼ + b
5/4 = b
The y-intercept is [tex](0, \frac{5}{4})[/tex] where [tex]b = \frac{5}{4}[/tex].
Linear Equation
The linear equation in slope-intercept form is [tex]y = \frac{1}{4}x + \frac{5}{4}[/tex] ⇒ Option F.
Verify the validity of other options:
Test Option C:
We must test whether Option C provides a true statement by substituting the value of [tex](1, \frac{3}{2})[/tex] into the equation.
[tex]y = \frac{1}{4}x + \frac{5}{4}[/tex]
[tex]\frac{3}{2} = \frac{1}{4}(1) + \frac{5}{4}[/tex]
[tex]\frac{3}{2} = \frac{1}{4} + \frac{5}{4}[/tex]
[tex]\frac{3}{2} = \frac{6}{4}[/tex]
[tex]\frac{3}{2} = \frac{3}{2}[/tex] ⇒ True statement.
Therefore, Option C is a valid answer.
Option A and B are also valid answers. We used the coordinates in Options A and B to solve for the slope.
Option D is an invalid answer, as there are more than 2 solutions to the given linear equation.
Option E is also a valid answer, as the points along the graph of a linear equation are solutions. Therefore, the given linear equation has infinitely many solutions.
Option G is also an invalid answer, as we have already determined that the linear equation that represents the given graph is represented by Option F.
Valid Answers:
Therefore, Options A, B, C, E, and F provide true statements about the given line.
