**TOPIC 10: COORDINATE OF A POINT **

Read the coordinates of a point

Coordinates of a points – are the values of 𝑥 and 𝑦 enclosed by the bracket which are used to describe the position of a point in the plane

The plane used is called 𝑥𝑦 − plane and it has two axis; horizontal axis known as 𝑥 − axis and; vertical axis known as 𝑦 − axis

**A Point Given its Coordinates**

Plot a point given its coordinates

Suppose you were told to locate (5, 2) on the plane. Where would you look? To understand the meaning of (5, 2), you have to know the following rule: Thex-coordinate (alwayscomes first. The first number (the first coordinate) isalwayson the horizontal axis.

**A Point on the Coordinates**

Locate a point on the coordinates

The location of (2,5) is shown on the coordinate grid below.

Thex-coordinate is 2. They-coordinate is 5. To locate (2,5), move 2 units to the right on thex-axis and 5 units up on they-axis.

The order in which you writex– andy-coordinates in an ordered pair is very important. Th ex-coordinate always comes first, followed by they-coordinate.

As you can see in the coordinate grid below, the ordered pairs (3,4) and (4,3) refer to two different points!

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**Gradient (Slope) of a Line**

The Gradient of a Line Given Two Points

Calculate the gradient of a line given two points

Gradient or slope of a line – is defined as the measure of steepness of the line. When using coordinates, gradient is defined as change in 𝑦 to the change in 𝑥.

Consider two points 𝐴 (𝑥1, 𝑦1)and (𝐵 𝑥2, 𝑦2), the slope between the two points is given by:

**Example 1**

Find the gradient of the lines joining:

(5, 1) and (2,−2)

(4,−2) and (−1, 0)

(−2,−3) and (−4,−7)

Solution

**Example 2**

The line joining (2,−3) and (𝑘, 5) has gradient −2. Find 𝑘

Find the value of 𝑚 if the line joining the points (−5,−3) and (6,𝑚) has a slope of½

Solution

**Equation of a Line**

The Equations of a Line Given the Coordinates of Two Points on a Line

Find the equations of a line given the coordinates of two points on a line

The equation of a straight line can be determined if one of the following is given:-

The gradient and the 𝑦 − intercept (at x = 0) or 𝑥 − intercept ( at y=0)

The gradient and a point on the line

Since only one point is given, then

Two points on the line

**Example 3**

Find the equation of the line with the following

Gradient 2 and 𝑦 − intercept −4

Gradient −2⁄3and passing through the point (2, 4)

Passing through the points (3, 4) and (4, 5)

Solution

Divide by the negative sign, (−), throughout the equation

∴The equation of the line is 2𝑥 + 3𝑦 − 16 = 0

The equation of a line can be expressed in two forms

𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and

𝑦 = 𝑚𝑥 + 𝑐

Consider the equation of the form 𝑦 = 𝑚𝑥 + 𝑐

𝑚 = Gradient of the line

**Example 4**

Find the gradient of the following lines

2𝑦 = 5𝑥 + 1

2𝑥 + 3𝑦 = 5

𝑥 + 𝑦 = 3

Solution

**Intercepts**

The line of the form 𝑦 = 𝑚𝑥 + 𝑐, crosses the 𝑦 − 𝑎𝑥𝑖𝑠 when 𝑥 = 0 and also crosses 𝑥 − 𝑎𝑥𝑖𝑠 when 𝑦 = 0

See the figure below

**Therefore**

to get 𝑥 − intercept, let 𝑦 = 0 and

to get 𝑦 − intercept, let 𝑥 = 0

From the line, 𝑦 = 𝑚𝑥 + 𝑐

𝑦 − intercept, let 𝑥 = 0

𝑦 = 𝑚 0 + 𝑐 = 0 + 𝑐 = 𝑐

𝑦 − intercept = c

Therefore, in the equation of the form 𝑦 = 𝑚𝑥 + 𝑐, 𝑚 is the gradient and 𝑐 is the 𝑦 − intercept

**Example 5**

Find the 𝑦 − intercepts of the following lines

Solution

**Graphs of Linear Equations**

The Table of Value

Form the table of value

**The graph of a straight line can be drawn by using two methods:**

i. By using intercepts

ii. By using the table of values

Example 6

Sketch the graph of 𝑦 = 2𝑥 − 1

Solution

**The Graph of a Linear Equation**

Draw the graph of a linear equation

By using the table of values

**Simultaneous Equations**

Linear Simultaneous Equations Graphically

Solve linear simultaneous equations graphically

Use the intercepts to plot the straight lines of the simultaneous equations. The point where the two lines cross each other is the solution to the simultaneous equations

**Example 7**

Solve the following simultaneous equations by graphical method

Solution

Consider: 𝑥 + 𝑦 = 4

If 𝑥 = 0, 0 + 𝑦 = 4 𝑦 = 4

If 𝑦 = 0, 𝑥 + 0 = 4 𝑥 = 4

Draw a straight line through the points 0, 4 and 4, 0 on the 𝑥𝑦 − plane

Consider: 2𝑥 − 𝑦 = 2

If 𝑥 = 0, 0 − 𝑦 = 2 𝑦 = −2

If 𝑦 = 0, 2𝑥 − 0 = 2 𝑥 = 1

Draw a straight line through the points (0,−2) and (1, 0) on the 𝑥𝑦 − plane

From the graph above the two lines meet at the point 2, 2 , therefore 𝑥 = 2 𝑎𝑛𝑑 𝑦 = 2

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