Transcript linear equation in one variable

Linear Equations and Inequalities
in One Variable
➋ • Linear Equations and Inequalities in One Variable
Linear equation in one variable
• A linear equation in one variable, also called a
first-degree equation in one variable, is one
where there is only one variable, and the
exponent of the variable is positive one (1).
• Ex. 3x  4  0
5  13 p  16
4a  5  7  3a
➋ • Linear Equations and Inequalities in One Variable
Solving an equation
• To solve an equation means to find the value
of the variable that will make the equation
true. And the value that will make the
equation true or will satisfy the equation is
called the solution.
• Illus. 9 is a solution of x – 5 = 4.
➋ • Linear Equations and Inequalities in One Variable
Properties of equality
• Reflexive Property: a = a
• Symmetric Property: a = b → b = a
• Transitive Property:
(a = b and b = c ) → a = c
• Addition Property of Equality:
a=b→a+c=b+c
• Multiplication Property of Equality:
a = b → ac = bc
➋ • Linear Equations and Inequalities in One Variable
• Ex.
5x  4  3x  6
0.17 x  0.07  0.03x
3(4  5 x)  2( x  4)  4
7 x  5  4x  9  10x  2  7 x  2
x 5 5x
 
1
2 3 4
➋ • Linear Equations and Inequalities in One Variable
Rational equations
• Equations which contain fractions that have
variables in the denominator are called
rational equations.
• Ex. x  7  3  9
7x
2x
14
1
x
4
1



2 x  2 2x  4 2
6x  6
2
3


x2  9 x  3 x  3
➋ • Linear Equations and Inequalities in One Variable
Literal equation
• To solve a literal equation means to derive a
formula for a certain unknown or variable
from a given formula.
• Ex. Solve C  95 ( F  32) for F .
n  nr 2
Solve A 
for n.
2
1 r
➋ • Linear Equations and Inequalities in One Variable
Word problems
• To solve a word problem in Algebra means to
find the value of an unknown quantity that
would satisfy the conditions stated in the
problem.
• Types of problems:
– Number relation
• Ex. Find the largest of four consecutive odd integers if
their sum is one hundred twenty.
➋ • Linear Equations and Inequalities in One Variable
Word problems
– Measurement
• Ex. The perimeter of a rectangle is eighty-six inches. If
the width of the rectangle is nineteen inches less than
the length, find the dimensions of the rectangle.
– Money
• Ex. Joseph invested Php1.4M in two banks, one giving
6% simple interest per annum and the other 7% simple
interest. After a year, the combined interest from both
banks is Php93,000. How much did he invest at 7%?
➋ • Linear Equations and Inequalities in One Variable
Word problems
– Work
• Ex. Bob can clean a classroom chalkboard in 30
seconds while Elmer can do the same job in 36
seconds. If they work together, how long will it take
them to clean a chalkboard?
– Mixture
• Ex. I have two kinds of Angel candy. The Cherub candy
sells at Php576 per kilo; the Seraph candy sells at
Php355 per kilo. If I mix 3.2 kilos of Cherub candies with
5.3 kilos of Seraph candies, how much should I sell the
mixed candies per kilo?
➋ • Linear Equations and Inequalities in One Variable
Inequalities
• Inequalities are mathematical statements that
use symbols, such as > , < , ≥ , and ≤.
• Two types:
– Absolute inequality, which is true for all values of
the variable(s)
• Illus. x  x  1
– Conditional inequality, which is true only for
certain values of the variable(s)
• Illus. y  2
➋ • Linear Equations and Inequalities in One Variable
Solving an inequality
• The solution to an inequality is a number or
set of numbers that, when substituted for the
variable, makes the inequality true. There are
three ways of showing the solution set,
namely, the graphical, interval, and setbuilder notations.
➋ • Linear Equations and Inequalities in One Variable
Ex.
3( x  2)  9
 5 x  14.5  0.5(9  2 x)
3  x 2x  4

4
3
5  3x  7  8
3  x  3x  1 and 3x  1  7 x  2
➋ • Linear Equations and Inequalities in One Variable
Word problems
• Ex. The circumference C of a circle with radius
r is computed using the formula C = 2πr. Find
the interval of values of the radius so that the
circumference is between 12 ft and 18 ft,
inclusive.
• Ex. A certain pediatric medicine must be
stored at a temperature between 5oC and
10oC. What is the equivalent temperature
range on the Fahrenheit scale?
➋ • Linear Equations and Inequalities in One Variable