Answer
The points that lie on the line are:
(4, 5) (OPTION A)
(10, 8) (OPTION C)
(20, 13) (OPTION D)
SOLUTION
Problem Statement
The question asks us which points lie on the line that passes through the points (0, 3) and (2, 4).
Method
- We are told that the line passes through the points (0,3) and (2, 4). This means that if we find the equation of the line connecting these two points, we can know if the other points we are asked about lie on the line as well.
- We will find the equation of the line using the formula given below:
[tex]\begin{gathered} \frac{y_2-y_1}{x_2-x_1}=\frac{y-y_1}{x-x_1} \\ \text{where,} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are the points connecting the line} \end{gathered}[/tex]
- The equation of the line can now be used to test if the points given lie on the line.
- We shall test the x-values of the individual points by plugging them into the equation. If they yield a y-value that corresponds to the y-value of the coordinate point we are testing, then, we can conclude that the point lies on the line.
- Thus, the steps we shall use to solve this question are simply:
1. Find the equation of the line.
2. Test the x-values of the points. That is: 4, 6, 10, 20, and 60.
Implementation
1. Find the Equation of the line:
[tex]\begin{gathered} \frac{y_2-y_1}{x_2-x_1}=\frac{y-y_1}{x-x_1} \\ \\ (x_1,y_1)=(0,3) \\ (x_2,y_2)=(2,4) \\ \\ \frac{4-3}{2-0}=\frac{y-3}{x-0} \\ \\ \frac{1}{2}=\frac{y-3}{x} \\ \\ \text{Cross multiply} \\ \frac{x}{2}=(y-3) \\ \\ \text{Add 3 to both sides} \\ y=\frac{x}{2}+3 \\ \\ \therefore\text{ the equation of the line is:} \\ y=\frac{x}{2}+3 \end{gathered}[/tex]
2. Test the x-values:
[tex]\begin{gathered} \text{ The values we will be testing are: 4, 6, 10, 20, and 60} \\ \\ \text{Testing 4:} \\ y=\frac{4}{2}+3=2+3=5 \\ \text{Thus, the point (4,5) lies on the line.} \\ \\ \text{Testing 6:} \\ y=\frac{6}{2}+3=3+3=6 \\ \text{Thus, the point (6,7) is NOT on the line.} \\ \\ \text{Testing 10:} \\ y=\frac{10}{2}+3=5+3=8 \\ \text{Thus, the point (10,8) lies on the line.} \\ \\ \text{Testing 20:} \\ y=\frac{20}{2}+3=10+3=13 \\ \text{Thus, the point (20,13) lies on the line.} \\ \\ \text{Testing 60:} \\ y=\frac{60}{2}+3=30+3=33 \\ \text{Thus, the point (60,34) is NOT on the line} \end{gathered}[/tex]
To confirm that these are the correct points that lie on the line, we can plot the points as follows
Final Answer
The points that lie on the line are:
(4, 5) (OPTION A)
(10, 8) (OPTION C)
(20, 13) (OPTION D)
