Topic 10: Coordinate Of a Point - Mathematics Form One

 

Welcome to our website darasahuru.co.tz, in this article, are you looking for Mathematics notes for Topic 10: Coordinate Of a Point - Mathematics Form One, Form one mathematics notes free download.


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 (always comes first. 


The first number (the first coordinate) is always on 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. The x-coordinate is 2. They-coordinate is 5. To locate (2,5), move 2 units to the right on the x-axis and 5 units up on they-axis.



The order in which you write x- and y-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!



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:


  1. (5, 1) and (2,−2)
  2. (4,−2) and (−1, 0)
  3. (−2,−3) and (−4,−7)


Solution



Example 2


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


  1. Gradient 2 and 𝑦 − intercept −4
  2. Gradient −2⁄3and passing through the point (2, 4)
  3. 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


  1. 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and
  2. 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


  1. 2𝑦 = 5𝑥 + 1
  2. 2𝑥 + 3𝑦 = 5
  3. 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


  1. to get 𝑥 − intercept, let 𝑦 = 0 and
  2. 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:


  1. By using intercepts
  2. 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
Previous Post Next Post

نموذج الاتصال