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References: Breadth, CGS System of Units, Metric System, SI Units.
LESS THAN (OR EQUAL TO)
References: Greater Than, Inequality.
LINE GRAPH 273
LIMITS OF ACCURACY
The line segment between A and B, the first line in the figure, is the set of points that move directly between A and B. The line segment AB is written as AB. The ray AB, the second line in the figure, is the infinite continuation^of the straight-line segment between A and B, in the direction of B. It is written as A?. The third line in the figure is the infinite continuation in both directions of the straight-line segment between A and B. It is written as AB.
A BA BA B
•------------------• •--------------------*-► -*-m------------------
Summary: The first figure is of a line segment, the second figure is of a ray, and the third figure is of a line.
References: Infinite, Vector.
This entry is confined to the statistical line graph, and is used when the data that are graphed are continuous. The following example explains how to draw and use a statistical line graph. For straight-line algebraic graphs see the entries Gradient-Intercept Form and Graphs.
Example. The temperature, in degrees Celsius, is recorded every 2 hours at East Cape Airport and the data are recorded in a table.
Time of day 2 am 4 am 6 am 8 am 10 am 12 noon 2 pm
Temperature f in degrees C 6 8 12 14 16 18 18
(a) Draw the line graph of these data.
(b) What is the temperature at 7 am?
Solution, (a) The graph is drawn with the time of day as the horizontal axis and the temperature as the vertical axis (see figure). The data from the table are plotted as the coordinates (2, 6), (4, 8), (6,12), (8, 14), (10, 16), (12, 18), and (14, 18), with 14 representing the time of 2 pm so as not to be confused with 2 am. Then the points are joined up with a series of straight-line segments. The result is a line graph. 0
(b) The temperature at 7 am is found by drawing a vertical line from the time axis to meet the graph, and from that point drawing a horizontal line to the temperature axis. The temperature at 7 am appears to be about 13°C.
References: Continuous Data, Gradient-Intercept Form, Graphs, Statistics.
LINE OF BEST FIT
References: Goodness of Fit, Scatter Diagram.
LINE OF SYMMETRY
References: Axis of Symmetry, Mirror Line.
This is an equation of degree one, and has one solution. A basic linear equation is of the form ax + b = c, where a, b, and c are constants. It may be necessary to read the entries Balancing an Equation and Inverse Operation before continuing. A practical introduction to solving a linear equation is explained in the following demonstration of how an equation is formed.
♦ Think of a number x
♦ Multiply it by 2 2x
♦ Add 3 to the result 2x + 3
♦ If the answer is 8, what is the number? Solve 2x + 3 = 8
LINEAR EQUATION 275
Starting with the answer, which is 8, we reverse the order of operations, and apply them as the appropriate inverse operations:
♦ Start with 8
♦ Subtract 3, which gives 5 “Subtract 3” is the inverse of “add 3”
♦ Divide by 2, which gives 2.5 “Divide by 2” is the inverse of “multiply by 2”
♦ The number thought of is 2.5
These ideas are formalized in the following example, using a technique known as “balancing an equation.” Whatever operation is done to one side of the equation must be done also to the other side to maintain a “balance.” Under this entry we study separately the various methods of solving linear equations using examples, and then solve equations involving more than one of these methods, including equations with brackets.
Method 1. Solve x — 8 = 2. Write
x — 8 + 8 = 2 + 8
Adding 8 to both sides of the equation reduces — 8 to zero on the left-hand side. Thus
x = 10
Method 2. Solve x + 3 = 2.9. Write
jc + 3 — 3 = 2.9 — 3
Subtracting 3 from both sides of the equation reduces +3 to zero on the left-hand side. Thus
x = —0.1
Method 3. Solve Ix = 19. Write
Ix _ 19
y _ y
Dividing both sides of the equation by 7 reduces 7 to 1 on the left-hand side. Thus x=2.1 19 = 7 = 2.7 (to 2 dp) or 2§
Method 4. Solve x/5 = 0.7. Note that x/5 is the same as Write
276 LINEAR EQUATION
Multiplying both sides of the equation by 5 reduces | to 1 on the left-hand side. Thus
We now solve equations that involve more than one of the above methods. When equations have more than one operation, as in those that follow, the order of operations for solving them is the reverse of the order of operations for forming the equations.
In the first example below the equation is formed by multiplying x by 3 and then dividing by 5. To solve the equation, we reverse the order and apply the inverse operations. The inverse of dividing by 5 is multiplying by 5. The inverse of multiplying by 3 is dividing by 3.