Question

question 1 of 10
which point lies on the line described by the equation below?
y + 4 = 4(x - 3)

a. (1, 11)
b. (2, 9)
c. (0, 0)
d. (3, -4)
e. (-2, 3)
f. (1, -11)

Explanation:

Step1: Recall the point - line relationship

A point \((x_0,y_0)\) lies on the line \(y - y_1=m(x - x_1)\) (point - slope form) if and only if when we substitute \(x = x_0\) and \(y=y_0\) into the equation, the equation holds true. The given equation is \(y + 4=4(x - 3)\), which can be written in point - slope form \(y-(-4)=4(x - 3)\), so the line passes through the point \((3,-4)\) (by the definition of point - slope form \(y - y_1=m(x - x_1)\) where \((x_1,y_1)\) is a point on the line and \(m\) is the slope). But we can also verify each option:

Step2: Verify option A \((1,11)\)

Substitute \(x = 1\) and \(y = 11\) into the left - hand side (LHS) and right - hand side (RHS) of the equation \(y + 4=4(x - 3)\).
LHS: \(y + 4=11 + 4=15\)
RHS: \(4(x - 3)=4\times(1 - 3)=4\times(-2)=-8\)
Since \(15
eq - 8\), the point \((1,11)\) does not lie on the line.

Step3: Verify option B \((2,9)\)

Substitute \(x = 2\) and \(y = 9\) into the equation.
LHS: \(y + 4=9 + 4 = 13\)
RHS: \(4(x - 3)=4\times(2 - 3)=4\times(-1)=-4\)
Since \(13
eq - 4\), the point \((2,9)\) does not lie on the line.

Step4: Verify option C \((0,0)\)

Substitute \(x = 0\) and \(y = 0\) into the equation.
LHS: \(y + 4=0 + 4 = 4\)
RHS: \(4(x - 3)=4\times(0 - 3)=4\times(-3)=-12\)
Since \(4
eq - 12\), the point \((0,0)\) does not lie on the line.

Step5: Verify option D \((3,-4)\)

Substitute \(x = 3\) and \(y=-4\) into the equation.
LHS: \(y + 4=-4 + 4 = 0\)
RHS: \(4(x - 3)=4\times(3 - 3)=4\times0 = 0\)
Since LHS = RHS, the point \((3,-4)\) lies on the line.

Step6: Verify option E \((-2,3)\)

Substitute \(x=-2\) and \(y = 3\) into the equation.
LHS: \(y + 4=3 + 4 = 7\)
RHS: \(4(x - 3)=4\times(-2 - 3)=4\times(-5)=-20\)
Since \(7
eq - 20\), the point \((-2,3)\) does not lie on the line.

Step7: Verify option F \((1,-11)\)

Substitute \(x = 1\) and \(y=-11\) into the equation.
LHS: \(y + 4=-11 + 4=-7\)
RHS: \(4(x - 3)=4\times(1 - 3)=4\times(-2)=-8\)
Since \(-7
eq - 8\), the point \((1,-11)\) does not lie on the line.

Answer:

D. (3, -4)