Question

which of these points lies on the line described by the equation below?
y - 3 = 5(x - 9)
a. (-3, -9)
b. (9, 3)
c. (3, 9)
d. (-9, -3)

Explanation:

Step1: Recall the point - line relationship

A point \((x,y)\) lies on the line \(y - y_1=m(x - x_1)\) (point - slope form) if substituting \(x\) and \(y\) into the equation satisfies it. The given equation is \(y - 3=5(x - 9)\). We will substitute the \(x\) and \(y\) values of each option into the equation to check.

Step2: Check option A \((-3,-9)\)

Substitute \(x=-3\) and \(y = - 9\) into the left - hand side (LHS) and right - hand side (RHS) of the equation.
LHS: \(y-3=-9 - 3=-12\)
RHS: \(5(x - 9)=5(-3 - 9)=5\times(-12)=-60\)
Since \(-12
eq - 60\), the point \((-3,-9)\) does not lie on the line.

Step3: Check option B \((9,3)\)

Substitute \(x = 9\) and \(y=3\) into the equation.
LHS: \(y - 3=3 - 3 = 0\)
RHS: \(5(x - 9)=5(9 - 9)=5\times0 = 0\)
Since LHS = RHS, the point \((9,3)\) satisfies the equation.

Step4: Check option C \((3,9)\) (for completeness)

Substitute \(x = 3\) and \(y = 9\) into the equation.
LHS: \(y-3=9 - 3=6\)
RHS: \(5(x - 9)=5(3 - 9)=5\times(-6)=-30\)
Since \(6
eq - 30\), the point \((3,9)\) does not lie on the line.

Step5: Check option D \((-9,-3)\) (for completeness)

Substitute \(x=-9\) and \(y=-3\) into the equation.
LHS: \(y - 3=-3 - 3=-6\)
RHS: \(5(x - 9)=5(-9 - 9)=5\times(-18)=-90\)
Since \(-6
eq - 90\), the point \((-9,-3)\) does not lie on the line.

Answer:

B. \((9,3)\)