Because the line passes through the origin, all of the points on the line will have an \(\displaystyle x:y\) ratio of \(\displaystyle 12:5\).  Only \(\displaystyle \left ( 15.6,8 \right )\) does not meet this requirement.

To find if a point lies on a graph or not, simply plug the x and y values into your equation and see if it holds true. Plugging our x values into the equation \(\displaystyle y=x^2-3x+7\) gives us the following:

\(\displaystyle x=1: y=(1)^2-3(1)+7=5\)

\(\displaystyle x=2: y=(2)^2-3(2)+7=5\)

\(\displaystyle x=3: y=(3)^2-3(3)+7=7\)

\(\displaystyle x=4: y=(4)^2-3(4)+7=11\) and

\(\displaystyle x=5: y=(5)^2-3(5)+7=17\)

So our points are \(\displaystyle (1,5), (2,5), (3,7), (4,11),\)and \(\displaystyle (5,17)\). This makes \(\displaystyle (4,12)\) the only point that does not lie on our graph. 

Substitute each answer choice into the equation in question, \(\displaystyle 2x-2y=1\), in order to test if the equation is valid at the given point

Only \(\displaystyle ( 1,\frac{1}{2})\) will satisfy the equation on both the left and right side.

The line \(\displaystyle x=1\) is a vertical line. This means that any point with an \(\displaystyle x\)-value of \(\displaystyle 1\) will exist on the line.

The only possible point with an \(\displaystyle x\)-value of \(\displaystyle 1\) is \(\displaystyle (1,20)\).

The line \(\displaystyle y=-1\) is a horizontal line with points all \(\displaystyle y\)-values equal to \(\displaystyle 1\).

The only possible answer with a \(\displaystyle y\)-value of \(\displaystyle -1\) is \(\displaystyle (1000,-1)\).

A point is on a line if, plugged in, it creates a true mathematical equation.

For example, \(\displaystyle (1,-2)\) is definitely on the line because when you plug it in, it creates a true equation:

\(\displaystyle y = 2x - 4\) plug in the \(\displaystyle x\) and \(\displaystyle y\) values

\(\displaystyle -2 = 2(1) - 4\) multiply

\(\displaystyle -2 = 2-4\)

\(\displaystyle -2 = -2\) this is true, so that point is on the line

The point that does not work is \(\displaystyle (-\frac{1}{2}, 5)\):

\(\displaystyle 5 = 2(-\frac{1}{2 }) -4\) multiply

\(\displaystyle 5 = -1 - 4\) subtract

\(\displaystyle 5 \neq -5\)

Since this is not true, the point is NOT on the line.

To find the solution to this problem just do a quick check, pluggin in each point's x and y value into the equation.  Remember coordinates are written with the x-value first, then the y-value (x,y).

If you try each point the only one that doesn't work is (8,2) !

\(\displaystyle y=\frac{1}{4} x + 2 = \frac{1}{4}(8) +2 = \frac{8}{4} +2 = 2 + 2 = 4 \neq 2\)

In other words, if the x-value is 8 the y-value should be 4!