To determine which point lies on the line described by the equation [tex]\( y + 3 = 2(x - 1) \)[/tex], let's first rewrite the equation in a more familiar form:
[tex]\[
y + 3 = 2(x - 1)
\][/tex]
Step 1: Distribute the 2 on the right side:
[tex]\[
y + 3 = 2x - 2
\][/tex]
Step 2: Subtract 3 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x - 2 - 3
\][/tex]
[tex]\[
y = 2x - 5
\][/tex]
The equation in slope-intercept form is:
[tex]\[
y = 2x - 5
\][/tex]
Now, we'll test each point to see if it satisfies this equation.
Point A: [tex]\((0,0)\)[/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the equation [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
0 = 2(0) - 5
\][/tex]
[tex]\[
0 = -5
\][/tex]
This is false. So, point [tex]\( (0, 0) \)[/tex] does not lie on the line.
Point B: [tex]\((-1,-6)\)[/tex]
Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -6 \)[/tex] into the equation [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
-6 = 2(-1) - 5
\][/tex]
[tex]\[
-6 = -2 - 5
\][/tex]
[tex]\[
-6 = -7
\][/tex]
This is false. So, point [tex]\( (-1, -6) \)[/tex] does not lie on the line.
Point C: [tex]\((1,-3)\)[/tex]
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
-3 = 2(1) - 5
\][/tex]
[tex]\[
-3 = 2 - 5
\][/tex]
[tex]\[
-3 = -3
\][/tex]
This is true. So, point [tex]\( (1, -3) \)[/tex] does lie on the line.
Point D: [tex]\((2,9)\)[/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 9 \)[/tex] into the equation [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
9 = 2(2) - 5
\][/tex]
[tex]\[
9 = 4 - 5
\][/tex]
[tex]\[
9 = -1
\][/tex]
This is false. So, point [tex]\( (2, 9) \)[/tex] does not lie on the line.
Point E: [tex]\((2,1)\)[/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the equation [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
1 = 2(2) - 5
\][/tex]
[tex]\[
1 = 4 - 5
\][/tex]
[tex]\[
1 = -1
\][/tex]
This is false. So, point [tex]\( (2, 1) \)[/tex] does not lie on the line.
Point F: [tex]\((1,-4)\)[/tex]
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -4 \)[/tex] into the equation [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
-4 = 2(1) - 5
\][/tex]
[tex]\[
-4 = 2 - 5
\][/tex]
[tex]\[
-4 = -3
\][/tex]
This is false. So, point [tex]\( (1, -4) \)[/tex] does not lie on the line.
Therefore, the point that lies on the line described by the equation [tex]\( y + 3 = 2(x - 1) \)[/tex] is:
[tex]\[ \boxed{(1, -3)} \][/tex]
