Answer :
To determine which of the given points lies on the graph of the equation \( y = 2x + 6 \), we will check each point by substituting its \( x \) and \( y \) values into the equation and seeing if the equation holds true.
Let's examine each point step-by-step:
Option A: (0, 8)
1. Substitute \( x = 0 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(0) + 6 = 0 + 6 = 6 \][/tex]
2. Check if \( y = 8 \):
[tex]\[ 6 \neq 8 \][/tex]
Thus, point (0, 8) does not lie on the graph.
Option B: (-1, 4)
1. Substitute \( x = -1 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(-1) + 6 = -2 + 6 = 4 \][/tex]
2. Check if \( y = 4 \):
[tex]\[ 4 = 4 \][/tex]
Thus, point (-1, 4) lies on the graph of the equation.
Option C: (1, 6)
1. Substitute \( x = 1 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(1) + 6 = 2 + 6 = 8 \][/tex]
2. Check if \( y = 6 \):
[tex]\[ 8 \neq 6 \][/tex]
Thus, point (1, 6) does not lie on the graph.
Option D: (2, -6)
1. Substitute \( x = 2 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(2) + 6 = 4 + 6 = 10 \][/tex]
2. Check if \( y = -6 \):
[tex]\[ 10 \neq -6 \][/tex]
Thus, point (2, -6) does not lie on the graph.
From this analysis, the point that lies on the graph of the equation \( y = 2x + 6 \) is:
[tex]\[ \boxed{(-1, 4)} \][/tex]
Therefore, the correct answer is option B: [tex]\((-1, 4)\)[/tex].
Let's examine each point step-by-step:
Option A: (0, 8)
1. Substitute \( x = 0 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(0) + 6 = 0 + 6 = 6 \][/tex]
2. Check if \( y = 8 \):
[tex]\[ 6 \neq 8 \][/tex]
Thus, point (0, 8) does not lie on the graph.
Option B: (-1, 4)
1. Substitute \( x = -1 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(-1) + 6 = -2 + 6 = 4 \][/tex]
2. Check if \( y = 4 \):
[tex]\[ 4 = 4 \][/tex]
Thus, point (-1, 4) lies on the graph of the equation.
Option C: (1, 6)
1. Substitute \( x = 1 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(1) + 6 = 2 + 6 = 8 \][/tex]
2. Check if \( y = 6 \):
[tex]\[ 8 \neq 6 \][/tex]
Thus, point (1, 6) does not lie on the graph.
Option D: (2, -6)
1. Substitute \( x = 2 \) into the equation \( y = 2x + 6 \):
[tex]\[ y = 2(2) + 6 = 4 + 6 = 10 \][/tex]
2. Check if \( y = -6 \):
[tex]\[ 10 \neq -6 \][/tex]
Thus, point (2, -6) does not lie on the graph.
From this analysis, the point that lies on the graph of the equation \( y = 2x + 6 \) is:
[tex]\[ \boxed{(-1, 4)} \][/tex]
Therefore, the correct answer is option B: [tex]\((-1, 4)\)[/tex].
