LINEAR RELATIONSHIPS AND BIVARIATE DATA
About "Linear relationships and bivariate data"
Linear relationships and bivariate data :
When data are collected in two variables simultaneously, say x and y, they are known as bivariate data.
The relationship represented by a bivariate data is linear, then its graph will be a line.
We can use the points on a graph of a linear relationship to write an equation for the relationship. The equation of a linear relationship is y = mx + b, where m is the rate of change, or slope, and b is the y-intercept (The value of y when x is 0).
Linear relationships and bivariate data - Examples
Example 1 :
A handrail runs alongside a stairway. As the horizontal distance from the bottom of the stairway changes, the height of the handrail changes. Show that the relationship is linear, and then find the equation for the relationship.
Solution :
Step 1 :
Connect the points in the graph given above.
Since all of the points (5, 7), (10, 11), (15, 15), (20, 19), and (25, 23) lie on the same line, the relationship is linear.
Step 2 :
Choose any two points in the form (x, y), from the graph to find the slope :
For example, let us choose (5, 7) and (10, 11).
Use the slope formula.
m = (y₂ - y₁) / (x₂ - x₁)
Substitute (5, 7) for (x₁, y₁) and (10, 11) for (x₂, y₂).
m = (11 - 7) / (10 - 5)
m = 4 / 5
m = 0.8
Step 3 :
Choose a point and use the slope to find the value of "b" (y-intercept b).
Substitute the point (5, 7) and m = 0.8 in the equation
y = mx + b
7 = (0.8)(5) + b
7 = 4 + b
3 = b
Step 4 :
To get the equation of the given linear relationship, plug m = 0.8 and b = 3 in the equation y = mx + b.
y = 0.8x + 3
Example 2 :
The table shows the relationship between time and cost. Show that the relationship is linear, and then find the equation for the relationship.
Solution :
Step 1 :
From the table, write the points in the form (x, y).
(5, 40), (10, 60), (20, 100) and (30, 140)
Step 2 :
Plot all the points in a graph paper and connect them.
Since all of the points (5, 40), (10, 60), (20, 100) and (30, 140) lie on the same line, the relationship is linear.
Step 3 :
Choose any two points in the form (x, y), from the graph to find the slope :
For example, let us choose (5, 40) and (10, 60).
Use the slope formula.
m = (y₂ - y₁) / (x₂ - x₁)
Substitute (5, 40) for (x₁, y₁) and (10, 60) for (x₂, y₂).
m = (60 - 40) / (10 - 5)
m = 20 / 5
m = 4
Step 4 :
Choose a point and use the slope to find the value of "b" (y-intercept b).
Substitute the point (5, 40) and m = 4 in the equation
y = mx + b
40 = (4)(5) + b
40 = 20 + b
20 = b
Step 5 :
To get the equation of the given linear relationship, plug m = 4 and b = 20 in the equation y = mx + b.
y = 4x + 20
After having gone through the stuff given above, we hope that the students would have understood "Linear relationships and bivariate data".
Apart from the stuff given above, if you want to know more about "Linear relationships and bivariate data", please click here
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Recent Articles
-
Point Symmetry Worksheet
Jun 05, 20 07:45 PM
Point Symmetry Worksheet
-
Point Symmetry
Jun 05, 20 07:34 PM
Point Symmetry - Concept - Example - Problems with step by step explanation
-
Symmetry
Jun 05, 20 07:18 PM
Symmetry - Concept - Examples
