Algebra Tutorial -straight Line

Algebra

Tutorial------

Straight

Lines

Straight lines are easy to lines are the most widely used

construct and use. figures.

Using graph paper [quadrille paper] you draw a horizontal line and You must have learnt all that. to draw any straight line.

we can draw a straight line…or else a vertical line cutting the horizontal line line. Here we use equations and show how

The equation for

In fact

Y =

straight line is this:

straight

a + bX

Here you find

two constants or numbers---- a' and 'b'. a' is called the Y intercept' or simply intercept. b' is called the slope' or gradient. For any straight line, we have to know these two things. Of course,some times, intercept may be zero or slope may be zero.But both cannot be zero! Let us practise

drawing a few lines

equation:

b=2 ON THE STRAIGHT LINE four points.and make a table: Y Straight Line y = 1 + 2x 0 1+0=1 5 1 1+2x1=3 4 2 1+2x2=5 3 -1 1+2x(-1)=-1 Y

1 Y= 1+ 2X Here you see a=1 Now let us find a few points Let us calculate the coordinates for X

using this

2

Let us just collect the numbers: 1 X Y 0 0 1 -1 1 3 -1 -0.5 2 5 -1 -1 Can you plot these four points using the X and y coordintes.Try this now.!

Y-Intercept

Line 2

0

0.5

1

1.5

2

X

We can plot and draw a straight line with just two points.But it is always better to pick three or four points, If you made anyeerror any error , it willinshow the calculation,it up in the graph. All th epoints will not lie on a single straight line! So, always do calculations three.or four points or at least for three points! Sometimes I am lazy and work out only This is the reason I did calculation for four points in the table above. Another simple suggestion: Pick points with x=0 ,x=1 and x= -1 These points are easy to calcu Can you see the Y-inercept in the graph above? Well, it is the line cuts the y axis. What do you see? Y=1 Yes, the Y inercept is: 1 From the equation, Y= 1 + 2X, you find that the intercept is Draw the line: Let us form

Y=3-X a small

Straight Line Y=3 - X

table:

4 3.5 3 2.5 Y

Note this:

Y

2 1.5

Y

Straight Line Y=3 - X 4 Y 3.5 -1 y=3-(-1)=4 3 0 y=3-0=3 2.5 1 y=3-1=2 2 2 y=3-2=1 3 y=3-3=0 1.5 Let us form the table again: 1 X Y 0.5 -1 4 0 0 3 -1 -0.5 0 0.5 1 1.5 2 2.5 3 1 2 X 2 1 3 0 For this line, I calculatedfor five points to illustrate! What is the value of the intercept? Y=3 When x=0, Y=3 That is the intercept and you see that in the table above.! Check with the figure! See the line Y=1+X above and this line Y=3-X now. What difference you find that strikes you first? Line Y=1+2X slopes upward wardor orYYvales values increase increase with x values with going increasing up. X values. Line Y= 3 - X slopes downward or y values decrease with x values going up. For line Y= 1+ 2X Slope is positive --> slope=2 For line Y= 3 - X, slope is negative--> slope= -1 Y

X

Slope

Y

Finding the slope from the points: Just pick two points and write down their coordinates: X Y x1--> 0 3 <----y1 x2--> 2 1 <----y2 slope= difference in Y/difference in X Slope = (y2-y1)/(x2-x1) Slope = (1 - 3)/(2 - 0) Slope= -1 From the equation Y = 3 + (-1)X b= -1 or slope =-1 Draw the line Y= -3 +2X The Y-inercept= -3 Slope= +2 Form the table: X Y -1 y=-3+2(-1)= -5 0 y=-3+2(0)=-3 1 y=-3 + 2(1)=-1 2 y=-3+2(2)=1 Wirte the table again X Y -1 -5 0 -3 1 -1 2 1

Straight Line Y=-3+2X 1 0.5 0 -0.5 -1 -1.5 Y

Line 3

-2

Y

-2.5 -3 -3.5 -4 -4.5 -5 -1

-0.5

0

0.5

1

1.5

X

Draw the line and intercept

with slope b= -2 a=-3

Straight Line Y= -3 - 2X -1 -1.5 -2 -2.5 -3 -3.5 Y

Line 4

-4 -4.5 -5

2

Straight Line Y= -3 - 2X

First write the equation:

-1

Y=a+bX Y= -3 -2 X

-1.5 -2 -2.5

X

-3

Y -1 0 1 2

-1 -3 -5 -7

-3.5 Y

Form the table:

-4 -4.5 -5 -5.5 -6

Draw the line

-6.5 -7 -1

-0.5

0

0.5

Summary

Well, these four

1

1.5

2

X

We have drawn

four straight lines 1. Positive Yintercept, 2 Positive intercept, 3 negative Y intercept, 4 negative Y intercept,

with Positive slope negative slope positive slope negative slope

lines represent

four kinds of straight lines you will

ever come across!

Point and slope form You are given one point on the straight line and its slope. say,

point P (2,1) lies on the line; Slope = 2 Can you construct the line. You know th e slope ,but not the inercept. Find the intercept first: From the equation Y=a+bX Put in y,x,slope=b

Then solve for the intercept 'a'

1= a + 2 x 2 1= a +4 1-4 = a +4 -4 a= -3 Now write the equation for the line: Proceed as before to draw the line.

Y = -3 +2X This is the line 4 given above.

Try this problem: point (2,-4) lies on the line . Slope = -1 Find the equation for the straight line and draw the line: Y = -4 = a -1x2 -1 Y= -4 = a -2 -1.25 a = -2 <--intercept -1.5 The equation is: Y= -2 -X -1.75 To draw the line, form the table: -2 X Y -2.25 -2.5 -1 -1 -2.75 0 -2 -3 1 -3 -3.25 2 -4 -3.5 Y

Straight Line

-3.75 -4 -1

-0.5

0

0.5 X

1

1.5

2

-2.75 -3 -3.25 -3.5 -3.75 -4 -1

-0.5

0

0.5

1

1.5

2

X

Two points form You are given

two points on the line:

(x1,y1)

and (x2,y2)

Slope = (y2-y1)/(x2-x1) Then take any one point and the slope obtained…calculate ed---Calculatethe theintercept intercept rcept. as given in the previous Then write the equation: Y=a+bX Example The two points lie on a straight line: (0,-2) andd (1,0) Find the equation of the line and plot it. Step1 Find

the slope: Slope = (0 - (-2))/ (1 -0) Slope= 2/1 = 2

Step2 Find the intercept

Take the point (0,-2) y=a+bX y=-2 = a + 2x0 y = -2 = a Step 3 Write the equation of the line: Y = -2 + 2X Step4 Draw the line 2 X Y 1.5 -1 -4 1 0.5 0 -2 0 -0.5 1 0 -1 2 2 -1.5 Y

Y =-2 + 2X

-2 -2.5 -3 -3.5 -4 -1

Do-it-Yourself

Exercises 1 Draw the line 2 Draw the line 3 Draw the line 4 Draw the line

0

0.5

1

1.5

2

X

Y = -3 +2.5X Y= 4 - 1.8X 2y = 4 + 3 X 3y = -9-12X

5 Draw the line passing through (2,1) 6 Draw the line passing through (-1,2) 7 Draw the line passing through (2,-3)

{hint: divide throughout by 2}

slope = 0.5 slope= 2.5 slope =2

8 Draw the line pasing through (-1,-1) and (2,2) 9 Draw th eline passing thorugh (-2,1) and (3,2) 10 Draw the line passing through (-2,-3) and (1,2) Additional Teasers 11 Draw the line passing through (0,0) and (2,2) 12 Draw the line : y= 3 {slope=0} 13 Draw the line : 2x = y +3 14 Draw the line : x=4

Applied Problems

-0.5

{What is the special feature of this line?

1 Johnny gets pocket money of $30 every month. He spends $2.25 every day for cookies and ice cream. Write the equation for Johnny's left-over money for any given day. After how many days, he will run out of pocket money? Draw the graph of money left over (y) and day of the month (x) 2 A train moves on a mountain track and climbs 55 meters run. If it starts at a height of 250 meters above sea-level, what would be its height after a run of 32 Km? Write the equation for height versus run and plot the graph. 3 A kingfisher is sitting on a cliff near the sea-shore at a height of 720 feet. a fish close to the surface of the water and swoops down at a speed of 120 feet per second.d Find the time when it will catch the fish. straight line path from cliff to the sea surface. 4 Sarah is organising the birthday party for her daughter. She spends $80 for the hall rent and $25 for the decorations of the hall. She would be spending $6 for food for each guest. Write the equation to express her total expense for the number of guests to be invited. Sarah wishes to limit her total expense to $300.00 How many guests she can invite so that she doesnot exceed her budget estimate of $300. Draw a graph of total expense [y] against number of guests [x]. 5 A tiger sees a deer, standing still, at a distance of 600 feet. the deer at a speed of 80 feet per second. After how many seconds, tiger will catch the deer,assuming , assuming the the deerdeer moves moves away away at aatspeed a speed of 30 of feet 30 meters per second. per second. [hint: the net speed = 80 -30 = 50 feet per second.] Parallel Lines have the same slope. Draw the lines y= 2 + 3x and The slopes are thesame: b= 3 Let us draw them in the same graph. X y=2+3x -1 0 1 2

Y= 3+3x The intercepts are different. Y= 3+3x -1 2 5 8

Parallel Lines

0 3 6 9

9 8 7 6 5 Y

Parallel Lines

4 3 2 1 0 -1 -1

-0.5

0

0.5

1

1.5

2

X

Perpendicular Lines

If two lines are perpendicular to each other,6 then their slopes are related: 5.5 If m amd m1 are the slopes of the two lines,then m.m1= -1 5 Take the pair of lines: y= 2 + 2x 4.5 Y=3-0.5X 4 Their slopes are 2 and -0.5; Their product3.5is -1. 3 y= 2+2x Let us draw them on the same graph: Y= 3 -0. 2.5 X y= 2+2x Y= 3 -0.5X 2 1.5 -1 0 3.5 Y

Perpendicular Lines

1 0.5 0 -1

-0.5

0

0.5 X

1

1.5

2

4.5 4

Y

3.5 3

y= 2+2x Y= 3 -0.

2.5 2 1.5

0 1 2

2 4 6

3 2.5 2

1 0.5 0 -1

-0.5

0

0.5 X

Note:

Please study the other "Algebra-Tutorials" written by me in Scribd.com Please give your feedback to improve these tutorials I welcome feedback from students, math teachers, and publishers. ********** **************** ************ ***********

1

1.5

2

both cannot be zero!

y = 1 + 2x

Y

1

1.5

2

ways better to pick

do calculations for es I am lazy and work out only three points.

hese points are easy to calculate--right! the point where

a=1

ht Line Y=3 - X

Y

ht Line Y=3 - X

Y

1

1.5

2

2.5

3

X

t in the table above.!

with increasing X values. x values going up.

and write down their

ght Line Y=-3+2X

Y

0.5

1

1.5

2

X

raight Line Y= -3 - 2X

Y

raight Line Y= -3 - 2X

Y

0

0.5

1

1.5

2

X

Straight Line

Y

0

0.5 X

1

1.5

2

0

0.5

1

1.5

2

X

Y =-2 + 2X

Y

0

0.5

1

1.5

2

X

he special feature of this line?}

nearly for any given day.

for every Km d be its height after a run ot the graph. It sees Draw the

She spends $80 for d be spending $6 for food total expense

exceed her budget er of guests [x].

It sprints towards many seconds, the 30 feet per second.

Parallel Lines

y=2+3x Y= 3+3x

-0.5

0

0.5

1

1.5

2

X

Perpendicular Lines

n their slopes are related:

-0.5

y= 2+2x Y= 3 -0.5X

0

0.5 X

1

1.5

2

-0.5

y= 2+2x Y= 3 -0.5X

0

0.5

1

1.5

X

parents

2