Points and Lines in the Coordinate Plane | Geometry | LevelKite

The coordinate plane lets you describe every location with numbers, so you can use algebra to answer geometry questions. It is a flat surface with two number lines that cross at right angles, at the 0 on each line. Each number line is called an axis.

The x-axis is the horizontal number line. Its positive numbers run to the right.
The y-axis is the vertical number line. Its positive numbers run upward.
The origin is the point where the axes cross.

Here is a helpful habit for this lesson: base every fact on a stated coordinate or a reason you can name. A figure only displays those facts, so never decide anything from how a picture looks.

Key concept

An ordered pair names one location

By the definition of the coordinate plane, each point has exactly one ordered pair $(x, y)$ of real numbers, and each ordered pair names exactly one point. The two numbers are the point's coordinates.

The x-coordinate comes first. It gives the point's distance from the y-axis, with a sign for the side: positive to the right, negative to the left, 0 on the axis.
The y-coordinate comes second. It gives the point's distance from the x-axis, with a sign for the side: positive above, negative below, 0 on the axis.

The graph of an equation is the set of all points whose ordered pairs make the equation true.

The axes split the rest of the plane into four regions called quadrants. They are numbered with the Roman numerals I, II, III and IV, going counterclockwise from the upper right. The table below defines each quadrant by the signs of the coordinates of its points. Every quadrant requires both coordinates to be nonzero, so a point on an axis lies in no quadrant.

In the figure below, the horizontal axis is the x-axis, the vertical axis is the y-axis, and each grid square is 1 unit wide. Each label gives that point's stated coordinates. The figure shows one point in each quadrant, one point on each axis, and the origin $(0, 0)$ , which lies on both axes.

The points

(3, 2)

,

(-4, 1)

,

(-2, -3)

and

(1, -4)

lie in Quadrants I, II, III and IV;

(5, 0)

is on the x-axis and

(0, 4)

is on the y-axis.

How the coordinates decide where a point lies

Location	x-coordinate	y-coordinate	Point from the figure
Quadrant I	$x > 0$	$y > 0$	$(3, 2)$
Quadrant II	$x < 0$	$y > 0$	$(-4, 1)$
Quadrant III	$x < 0$	$y < 0$	$(-2, -3)$
Quadrant IV	$x > 0$	$y < 0$	$(1, -4)$
x-axis	any real number	$y = 0$	$(5, 0)$
y-axis	$x = 0$	any real number	$(0, 4)$

Worked example

Find where each point lies

Given the points $(-3, 5)$ , $(4, -2)$ , $\left(-\tfrac{1}{2}, -\tfrac{7}{2}\right)$ and $(0, -6)$ , find where each one lies.

Start with (-3, 5). Its x-coordinate is negative and its y-coordinate is positive. Those are exactly the signs in the definition of Quadrant II, so the point lies in Quadrant II.
$x = -3 < 0, \quad y = 5 > 0$
Apply the same test to (4, -2). Its signs match the definition of Quadrant IV.
$x = 4 > 0, \quad y = -2 < 0$
The third point has fractions for coordinates, but the test uses only signs. Both coordinates are negative, which matches the definition of Quadrant III.
$x = -\tfrac{1}{2} < 0, \quad y = -\tfrac{7}{2} < 0$
The last point has x-coordinate 0. By the definition of the x-coordinate, it lies on the y-axis. A quadrant requires both coordinates to be nonzero, so this point is in no quadrant.
$x = 0, \quad y = -6$

Answer

The point $(-3, 5)$ lies in Quadrant II, $(4, -2)$ lies in Quadrant IV, $\left(-\tfrac{1}{2}, -\tfrac{7}{2}\right)$ lies in Quadrant III, and $(0, -6)$ lies on the y-axis.

To graph a point means to mark the one location its ordered pair names. Start at the origin, where both coordinates are 0. Move left or right to set the x-coordinate, then move up or down to set the y-coordinate. This works because a left-or-right move does not change your distance from the x-axis, and an up-or-down move does not change your distance from the y-axis. A coordinate of 0 means no move in that direction.

Keep coordinates exact: leave fractions as fractions and radicals in simplest form. If you use a decimal to help place a point, label it as an approximation, as in $\sqrt{2} \approx 1.41$ .

Worked example

Graph two points

Given Point A at $(-4, 3)$ and Point B at $\left(\tfrac{5}{2}, -2\right)$ , graph both points.

For Point A, start at the origin. Its x-coordinate is -4, and by the definition of the x-coordinate a negative value means left of the y-axis. Move 4 units left.
$(0, 0) \to (-4, 0)$
Its y-coordinate is 3, and by the definition of the y-coordinate a positive value means above the x-axis. Move 3 units up and mark Point A.
$(-4, 0) \to (-4, 3)$
For Point B, start at the origin again. Its x-coordinate is 5/2, which equals 2 + 1/2, so move right and stop exactly halfway between 2 and 3.
$(0, 0) \to \left(\tfrac{5}{2}, 0\right)$
Its y-coordinate is -2, so move 2 units down and mark Point B.
$\left(\tfrac{5}{2}, 0\right) \to \left(\tfrac{5}{2}, -2\right)$

Answer

Point A is graphed 4 units left of the y-axis and 3 units above the x-axis, in Quadrant II. Point B is graphed $\tfrac{5}{2}$ units right of the y-axis and 2 units below the x-axis, in Quadrant IV.

Point A is at

(-4, 3)

, Point B is at

\left(\tfrac{5}{2}, -2\right)

, and Point C is at the swapped pair

(3, -4)

. The next example names Points D and E.

Now go the other way, from a point to its ordered pair. To name the coordinates of a point, you need its distance and direction from each axis. A problem may state these in words. A figure can supply them too, but only when it is stated that the point sits at a grid crossing and how wide the grid squares are. Then counting squares gives exact distances. Without those statements, a picture only suggests a location, so you cannot claim exact coordinates from it.

In the figure above, the horizontal axis is the x-axis, the vertical axis is the y-axis, and each grid square is 1 unit wide.

Worked example

Name the coordinates of two points

Given: in the figure above, Point D sits at a grid crossing 5 squares left of the y-axis and 2 squares below the x-axis, and Point E sits at a grid crossing on the y-axis, 4 squares above the x-axis. Each square is 1 unit wide. Name the coordinates of each point.

Each square is 1 unit wide, so Point D is 5 units left of the y-axis. By the definition of the x-coordinate, left means negative.
$x = -5$
Point D is 2 units below the x-axis. By the definition of the y-coordinate, below means negative.
$y = -2$
Write the x-coordinate first, as the definition of an ordered pair requires. As a check, the two negative signs match the definition of Quadrant III, the region left of the y-axis and below the x-axis.
$D = (-5, -2)$
Point E is on the y-axis, so its x-coordinate is 0. It is 4 units above the x-axis, so its y-coordinate is 4.
$E = (0, 4)$

Answer

Point D has coordinates $(-5, -2)$ and Point E has coordinates $(0, 4)$ .

An equation in two variables is true for some ordered pairs and false for others. A pair that makes it true is a solution. For example, $(3, 0)$ is a solution of $2x + 3y = 6$ because $2(3) + 3(0) = 6$ . The pair $(1, 1)$ is not a solution because $2(1) + 3(1) = 5$ and $5 \neq 6$ . By the definition of a graph, $(3, 0)$ is on the graph of this equation and $(1, 1)$ is not.

A linear equation in two variables is an equation that can be written as $ax + by = c$ , where $a$ , $b$ and $c$ are real numbers and $a$ and $b$ are not both 0. For example, $y = 2x - 1$ is linear: subtracting $2x$ from both sides (Subtraction Property of Equality) gives $-2x + y = -1$ .

A postulate is a statement accepted without proof, and a theorem is a statement that has been proved. Graphing a linear equation rests on one of each.

Linear Graph Theorem: if an equation can be written as $ax + by = c$ with $a$ and $b$ not both 0, then its graph is a line. You will use this theorem without seeing its proof here. Its converse, which swaps the if and then parts, says that if a figure is a line, then it is the graph of such an equation. The converse is also true, but here you use only the first direction: from equation to line.
Two-Point Postulate: through any two distinct (different) points there is exactly one line.

So your plan is to find two distinct solutions, plot them, and draw the one line through them. Then find a third different solution to guard against a plotting mistake.

Worked example

Graph a linear equation

Given the equation $2x + 3y = 6$ , graph it. Here $a = 2$ , $b = 3$ and $c = 6$ , so $a$ and $b$ are not both 0 and the Linear Graph Theorem applies.

Choose x = 0, because that removes the x term. Substitute 0 for x (Substitution Property of Equality) and simplify. Then divide both sides by 3; the Division Property of Equality allows this because 3 is not 0. This gives the solution (0, 2).
$2(0) + 3y = 6 \;\Rightarrow\; 3y = 6 \;\Rightarrow\; y = \tfrac{6}{3} = 2$
Choose y = 0 and repeat: substitute, simplify, and divide both sides by 2, which is not 0. This gives the solution (3, 0), a different point from (0, 2).
$2x + 3(0) = 6 \;\Rightarrow\; 2x = 6 \;\Rightarrow\; x = \tfrac{6}{2} = 3$
Choose x = -3 for a third solution. Substitute and multiply, add 6 to both sides (Addition Property of Equality), then divide both sides by 3. This gives (-3, 4).
$2(-3) + 3y = 6 \;\Rightarrow\; -6 + 3y = 6 \;\Rightarrow\; 3y = 12 \;\Rightarrow\; y = \tfrac{12}{3} = 4$
Check all three pairs in the original equation. Each one gives a true statement, so by the definition of a solution, each pair is a solution.
$2(0) + 3(2) = 6, \quad 2(3) + 3(0) = 6, \quad 2(-3) + 3(4) = 6$
Plot (0, 2), (3, 0) and (-3, 4) in the figure below, where the horizontal axis is the x-axis, the vertical axis is the y-axis, and each grid square is 1 unit wide. By the Linear Graph Theorem the graph is a line, and by the Two-Point Postulate exactly one line passes through (0, 2) and (3, 0). Draw that line and extend it in both directions.
By the definition of a graph, the third solution must also lie on this line. If your plotted (-3, 4) misses your line, you have made a mistake, so recheck your work.

Answer

The graph of $2x + 3y = 6$ is the one line through $(0, 2)$ and $(3, 0)$ , and that line also contains $(-3, 4)$ .

Line A is

2x + 3y = 6

, through the solutions

(0, 2)

,

(3, 0)

and

(-3, 4)

. Line B is

x = -5

and Line C is

y = -3

, explained next.

Setting $x = 0$ and then $y = 0$ gives two distinct solutions only when $a$ , $b$ and $c$ are all nonzero. In the other cases, you still need two distinct solutions, so choose your values differently.

If $c = 0$ and $a$ and $b$ are nonzero, both choices give the origin, which is only one point. Choose another value, such as $x = 1$ .
If $b = 0$ , then $a \neq 0$ and the equation becomes $ax = c$ . Dividing by $a$ gives $x = \tfrac{c}{a}$ , and $y$ can be any real number, so the graph is a vertical line. Line B is $x = -5$ , which is $1x + 0y = -5$ . It contains $(-5, 0)$ and $(-5, 2)$ .
If $a = 0$ , then $b \neq 0$ and the equation becomes $by = c$ . Dividing by $b$ gives $y = \tfrac{c}{b}$ , and $x$ can be any real number, so the graph is a horizontal line. Line C is $y = -3$ , which is $0x + 1y = -3$ . It contains $(0, -3)$ and $(2, -3)$ .

Takeaway: an ordered pair $(x, y)$ names one exact location, with $x$ first and $y$ second. The signs of the coordinates give the quadrant, and a coordinate of 0 puts the point on an axis. The graph of a linear equation is a line made of all its solutions, so any two distinct solutions determine the whole graph.