PSAT Math - How to find out if a point is on a line with an equation

Points D and E lie on the same line and have the coordinates $\left ( 1,2\right )$ and $\left ( 5,7\right )$ , respectively. Which of the following points lies on the same line as points D and E?

$\left ( 3,4.5\right )$

$\left ( 2,3.5\right )$

$\left ( 2,4\right )$

$\left ( 3,3.25\right )$

$\left ( 3,4\right )$

Explanation

The first step is to find the equation of the line that the original points, D and E, are on. You have two points, so you can figure out the slope of the line by plugging the points into the equation

$slope = \frac{\Delta y}{\Delta x} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ .

$slope = \frac{7-2}{5-1} = \frac{5}{4}$

Therefore, you can get an equation in the line in point-slope form, which is

Plug in the answer options, and you will find that only the point $\left ( 3,4.5\right )$ solves the equation.

Which of the following points is on the line given by the equation $y=\frac{1}{2}x+3$ ?

Explanation

In order to solve this, try each of the answer choices in the equation:

$y=\frac{1}{2}x+3$

For example, when we try (3,4), we find:

$4=\frac{1}{2}(3)+3$

This does not work. When we try all the choices, we find that only (2,4) works:

$y=\frac{1}{2}x+3$

$4=\frac{1}{2}(2)+3$

Which of the following lines contains the point (8, 9)?

$\dpi{100} \small 3x-6=2y$

$\dpi{100} \small 8x=9y$

$\dpi{100} \small 8x+9=y$

$\dpi{100} \small 3x+6=2y$

$\dpi{100} \small 3x+6=y$

Explanation

In order to find out which of these lines is correct, we simply plug in the values $\dpi{100} \small x=8$ and $\dpi{100} \small y=9$ into each equation and see if it balances.

The only one for which this will work is $\dpi{100} \small 3x-6=2y$

The equation of a line is: 2x + 9y = 71

Which of these points is on that line?

(4,7)

(4,-7)

(-4,7)

(2,7)

(-2,7)

Explanation

Test the difference combinations out starting with the most repeated number. In this case, y = 7 appears most often in the answers. Plug in y=7 and solve for x. If the answer does not appear on the list, solve for the next most common coordinate.

2(x) + 9(7) = 71

2x + 63 = 71

2x = 8

x = 4

Therefore the answer is (4, 7)

$\dpi{100} \small 5x+25y = 125$

Which point lies on this line?

$\dpi{100} \small (5,4)$

$\dpi{100} \small (1,5)$

$\dpi{100} \small (5,1)$

$\dpi{100} \small (5,5)$

$\dpi{100} \small (1,4)$

Explanation

$\dpi{100} \small 5x+25y = 125$

Test the coordinates to find the ordered pair that makes the equation of the line true:

$\dpi{100} \small (5,4)$

$\dpi{100} \small 5 (5) + 25 (4) = 25 + 100 = 125$

$\dpi{100} \small (1,5)$

$\dpi{100} \small 5(1)+25(5)= 5+125=130$

$\dpi{100} \small (5,1)$

$\dpi{100} \small 5(5)+25(1)= 25+25=50$

$\dpi{100} \small (5,5)$

$\dpi{100} \small 5(5)+25(5)= 25+125=150$

$\dpi{100} \small (1,4)$

$\dpi{100} \small 5(1)+25(4)= 5+100=105$

In the xy -plane, line l is given by the equation 2_x_ - 3_y_ = 5. If line l passes through the point (a ,1), what is the value of a ?

-1

-2

Explanation

The equation of line l relates x -values and y -values that lie along the line. The question is asking for the x -value of a point on the line whose y -value is 1, so we are looking for the x -value on the line when the y-value is 1. In the equation of the line, plug 1 in for y and solve for x:

2_x_ - 3(1) = 5

2_x_ - 3 = 5

2_x_ = 8

x = 4. So the missing x-value on line l is 4.

How to find out if a point is on a line with an equation

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