Equation of a Straight Line

Equation of a Straight Line

Introduction
Equation of a straight line is a useful concept in daily life, as it is used in different fields; for example, it is used in business when predicting profits, in production when calculating wages basing on an hourly pay rate, and in medicine when calculating medicine doses basing on patients’ weights. In order to appreciate more applications of equations of straight lines in real life, you need to understand and use linear equations and their graphs.

1.1 Understanding the Relationship between a Linear
Equation y = mx + c and its Graph
In Senior One, you learnt how to form linear equations using given points and drawing the graph of a line given the equation. You are going to use that background to understand the relationship between a linear equation and its graph.

Reviewing the linear equation y = mx + c
Activity 1.1 (a)
Suppose that you and your friend are planning to harvest beans. It is known that you can hire workers (x) to help you in harvesting beans (y), and the quantity of beans harvested (in kg) for each number of workers is as planned in the tables below.

(a)Study and write down the set of points in each of the tables above.
Form a linear equation for each set of points in (a).
Comment on the two equations formed in (b).
The x- and y-coordinates of the point (x, y) which lies on a straight line,
have the linear relationship y = mx + c.

The relationship between the linear equation y = mx + c and its graph
Under this section, you will understand the relationship between the linear equation
y = mx + c and its graph.

Activity 1.1 (c)
(Work in groups)
Suggested materials:

• a foot ruler (30 cm ruler)
• a protractor
  Instructions:
• a desk/ table
  . a pencil
  (a) Place the foot of the ruler at 7 cm from the desk/table and lean the ruler against the side of a desk/table.
  (b) Measure the angle between the floor and foot of the ruler.
  (c) Copy and complete the following table by repeating procedures (a) and (b).

ICT Activity
Suggested material: Internet-enabled device
In groups:
(a) Using the internet, go to the link: mathsisfun.com/data/straight line grqphåtml
(b) Study the effect of the gradient and y-intercept on the graph by changing the
values of the gradient and y-intercept.
(c) Explain your observations to the class.

1.2 Determining the x- and y-Intercepts of a given Linear Graph
In Activity 7.7(c), you observed that the distance between the foot of the ruler and the side of the desk determined how far from the side of the desk the ruler leaned. Taking the ruler as a straight line, one of its ends on the floor and the side of the desk demonstrate the x- and y- intercepts, respectively.

In the linear equation y = mx + c, c is called the y-intercept. The y-intercept is the y-coordinate of the point where the graph of the line intersects they-axis (or the line x = 0).

1.3 Determining the Gradient of a Straight Line
In the graph of a line y = mx + c, the value of m describes the steepness of the graph.
Activity 1.3(a) (Work in groups)
Suggested materials:

• a graph paper
• a pencil
• a ruler

Instructions:
(a) Plot the points A(-3, -2), B(-l, 2), and C(2, 8) on a graph paper.
(b) Join the points you have plotted above, to form a straight line,
(c) Find the change in x values and that iny values for each pair of points.
(d) Calculate the ratio of the change iny values to the change in the x values for
each pair of points in (c). Comment on your results.
(e) Hence, state the gradient of the straight line formed.

1.4 Stating the Gradient of a Straight Line when given the Equation
Activity 1.4(a) (Work in groups)
(a) State the general form of equation of a straight line.
(b) Describe all the letters in that equation.
(c) Present your work to the class.
Activity 1.4(b) (Work in groups)
Given the equations 2y = 4x + 7 and 3y + 2x = 5;
(a) re-arrange each equation to the general form of the equation of a straight line.
(b) state the gradient of each line.
(c) compare your results with other groups.

1.5 Applying the Relationships of Gradients of Parallel and Perpendicular Lines to Determine the Equation of a Straight Line
The gradients of parallel lines have a unique relationship. The same applies to perpendicular lines.
Applying the relationship between the gradients of parallel lines
Activity 1.5(a)
(a) Draw three parallel lines of your choice.
(b) Describe the features of the parallel lines drawn.
Activity 1.5(b)
(a) On the same axes, plot the following lines.
(b)Describe the features of the lines in (a).
(c)Use your graphs to state the intercepts of the lines.
(d)Find the gradient of each line. What do you observe?
(e)Comment on your observations.
iii)

Applying the relationship between the gradients of perpendicular lines
Activity 1.5(c) (Work in groups)
(a)Identify perpendicular lines in your classroom environment.
(b)Draw perpendicular lines and state the conditions for a line to be perpendicular to another
(c)Draw the graphs of the lines 5y = 4x + 10 and 4y+ 5x = 4 on the same pair of axes.
(d)Where do the two lines intersect?
(e)State the gradient of each of the lines.
(f) Discuss how the gradients of the two lines are related.

ICT Activity
Identify where the concept of straight lines is applicable in your society. Take necessary photographs regarding your identification and present your work to the class.