Answer :
To determine which point lies on the line given by the point-slope equation [tex]\( y + 5 = 2(x + 8) \)[/tex], we will convert this equation into its slope-intercept form and then check each point to see if it satisfies the equation.
First, let's rearrange the given equation to get it into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y + 5 = 2(x + 8) \][/tex]
Subtract 5 from both sides:
[tex]\[ y = 2(x + 8) - 5 \][/tex]
Now, distribute the 2 on the right-hand side:
[tex]\[ y = 2x + 16 - 5 \][/tex]
Simplify the expression:
[tex]\[ y = 2x + 11 \][/tex]
Now, we have the equation of the line in slope-intercept form: [tex]\( y = 2x + 11 \)[/tex]. Next, we'll substitute the coordinates of each point into the equation [tex]\( y = 2x + 11 \)[/tex] to see if they satisfy the equation.
Option A: [tex]\((-8, -5)\)[/tex]
Substitute [tex]\( x = -8 \)[/tex] and [tex]\( y = -5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ -5 = 2(-8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ -5 = -16 + 11 \][/tex]
[tex]\[ -5 = -5 \][/tex]
This point satisfies the equation.
Option B: [tex]\((8, -5)\)[/tex]
Substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = -5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ -5 = 2(8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ -5 = 16 + 11 \][/tex]
[tex]\[ -5 = 27 \][/tex]
This point does not satisfy the equation.
Option C: [tex]\((8, 5)\)[/tex]
Substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = 5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ 5 = 2(8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ 5 = 16 + 11 \][/tex]
[tex]\[ 5 = 27 \][/tex]
This point does not satisfy the equation.
Option D: [tex]\((-8, 5)\)[/tex]
Substitute [tex]\( x = -8 \)[/tex] and [tex]\( y = 5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ 5 = 2(-8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ 5 = -16 + 11 \][/tex]
[tex]\[ 5 = -5 \][/tex]
This point does not satisfy the equation.
After checking all the points, we find that only point A [tex]\((-8, -5)\)[/tex] satisfies the equation [tex]\( y + 5 = 2(x + 8) \)[/tex]. Therefore, the point that lies on the line is:
A. [tex]\((-8, -5)\)[/tex]
First, let's rearrange the given equation to get it into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y + 5 = 2(x + 8) \][/tex]
Subtract 5 from both sides:
[tex]\[ y = 2(x + 8) - 5 \][/tex]
Now, distribute the 2 on the right-hand side:
[tex]\[ y = 2x + 16 - 5 \][/tex]
Simplify the expression:
[tex]\[ y = 2x + 11 \][/tex]
Now, we have the equation of the line in slope-intercept form: [tex]\( y = 2x + 11 \)[/tex]. Next, we'll substitute the coordinates of each point into the equation [tex]\( y = 2x + 11 \)[/tex] to see if they satisfy the equation.
Option A: [tex]\((-8, -5)\)[/tex]
Substitute [tex]\( x = -8 \)[/tex] and [tex]\( y = -5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ -5 = 2(-8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ -5 = -16 + 11 \][/tex]
[tex]\[ -5 = -5 \][/tex]
This point satisfies the equation.
Option B: [tex]\((8, -5)\)[/tex]
Substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = -5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ -5 = 2(8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ -5 = 16 + 11 \][/tex]
[tex]\[ -5 = 27 \][/tex]
This point does not satisfy the equation.
Option C: [tex]\((8, 5)\)[/tex]
Substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = 5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ 5 = 2(8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ 5 = 16 + 11 \][/tex]
[tex]\[ 5 = 27 \][/tex]
This point does not satisfy the equation.
Option D: [tex]\((-8, 5)\)[/tex]
Substitute [tex]\( x = -8 \)[/tex] and [tex]\( y = 5 \)[/tex] into [tex]\( y = 2x + 11 \)[/tex]:
[tex]\[ 5 = 2(-8) + 11 \][/tex]
Evaluate the right side:
[tex]\[ 5 = -16 + 11 \][/tex]
[tex]\[ 5 = -5 \][/tex]
This point does not satisfy the equation.
After checking all the points, we find that only point A [tex]\((-8, -5)\)[/tex] satisfies the equation [tex]\( y + 5 = 2(x + 8) \)[/tex]. Therefore, the point that lies on the line is:
A. [tex]\((-8, -5)\)[/tex]
