Answer:
We conclude that only the equation y = 3x-11 is not an equation of the line that passes through the points.
Hence, option B is correct.
Step-by-step explanation:
Given the points
Finding the slope between (2, -1) and (4, 5)
[tex]\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\left(x_1,\:y_1\right)=\left(2,\:-1\right),\:\left(x_2,\:y_2\right)=\left(4,\:5\right)[/tex]
[tex]m=\frac{5-\left(-1\right)}{4-2}[/tex]
[tex]m=3[/tex]
Using the point-slope form of the line equation
[tex]y-y_1=m\left(x-x_1\right)[/tex]
where m is the slope of the line and (x₁, y₁) is the point
substituting the values m = 3 and the point (2, -1)
[tex]y-y_1=m\left(x-x_1\right)[/tex]
[tex]y-2=3\left(x-\left(-1\right)\right)[/tex]
[tex]y-2=3\left(x+1\right)[/tex]
Add 2 to both sides
[tex]y-2+2=3\left(x+1\right)+2[/tex]
[tex]y=3x+5[/tex]
So, y = 3x+5 is one of the equation lines.
We can also check by substituting the values m = 3 and the point (4, 5)
[tex]y-y_1=m\left(x-x_1\right)[/tex]
y - 5 = 3(x-4)
So, y - 5 = 3(x-4) is also one of the equation lines.
Now, let say take the point (4, 5) an substitute in y = 3x-11
y = 3x-11
put x = 4 and y = 5
5 = 3(4)-11
5 = 12-11
5 = 1
FALSE!
Hence, the equation y = 3x-11 does not pass through (4, 5). Thus, the equation y = 3x-11 is not an equation of the line that passes through the points.
Now, let say take the point (4, 5) an substitute in y+1 = 3(x-2)
y+1 = 3(x-2)
put x = 4 and y = 5
5+1 = 3(4-2)
6 = 3(2)
6 = 6
TRUE!
Now, let say take the point (2, -1) an substitute in y+1 = 3(x-2)
-1+1 = 3(2-2)
0 = 0
TRUE!
Thus, the equation y+1 = 3(x-2) is an equation of the line that passes through the points.
Now, let say take the point (2, -1) an substitute in y = 3x-7
y = 3x-7
put x = 2 and y = -1
-1 = 3(2) - 7
-1 = 6-7
-1 = -1
TRUE!
Now, let say take the point (4, 5) an substitute in y = 3x-7
y = 3x-7
put x = 4 and y = 5
5 = 3(4)-7
5 = 12-7
5 = 5
TRUE!
Thus, the equation y = 3x-7 is an equation of the line that passes through the points.
Conclusion:
We conclude that only the equation y = 3x-11 is not an equation of the line that passes through the points.
Hence, option B is correct.
