The letters x and y are often used as variables, although any letter can be used.
An algebraic expression has one or more variables and can be written as a single term or as a sum of terms. Here are some examples of algebraic expressions. 2 x , w^3 z + 5 z^2 - z^2 + 6
A term that has no variable is called a constant term.
A number that is multiplied by variables is called the coefficient of a term. For example, in the expression 2 x 2 + 7 x - 5, 2 is the coefficient of the term 2 x 2 , 7 is the coefficient of the term 7 x, and -5 is a constant term.
Here are some standard identities that are useful.
( a + b)^2 = a2 + 2ab + b 2 ( a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 a^2 - b^2 = (a + b)(a - b)
In the algebraic expression x^a ,
where x is raised to the power a, x is called the base and a is called the exponent.
If x^a = x^b , then a = b.
This is true for all positive numbers x, except x = 1, and for all integers a and b.
ex: 4^-3= 1/4^3
(x^a)(x^b) = x^(a+b)
5^7 / 5^2= 5
x^0 = 1
(x^a)(y^a) = (x*y)^a
2^3*4^3=8^3 ; (10Z)^3 = 10^3 Z^3= 1000 Z^3
(x/y)^a = (x^a) / (y^2)
(r/4t)^3 = r^3 / 4t^3
((x)^a)^b = x^a*b
(2^2)^3 = 2^6
Linear Equations A linear equation in two variables, x and y, can be written in the form
where a, b, and c are real numbers and a and b are not both zero. For example, 3 x + 2 y = 8 is a linear equation in two variables.
to solve the system of equations 4 x + 3 y = 13 x + 2 y = 2
4 (2 - 2 y ) + 3 y = 13 8 - 8 y + 3 y = 13 => -8 y + 3 y = 5 => -5 y = 5 => y = -1 x + 2 (-1) = 2 = > x = 4
The following rules are important for producing equivalent equations.
• When the same constant is added to or subtracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original equation. • When both sides of an equation are multiplied or divided by the same nonzero constant, the equality is preserved and the new equation is equivalent to the original equation.
Quadratic Equations معادلات درجه دوم
In the quadratic equation 2 x^ 2 - x - 6 = 0 , we have a = 2, b = -1, and c = - 6. Therefore, the quadratic formula yields x1=2 , x2=-3/2
If the square of the number x is multiplied by 3, and then 10 is added to that product,
the result can be represented by 3x^2 + 10.
If John's present salary s is increased by 14 percent,
then his new salary is 1.14s
If y gallons of syrup are to be distributed among 5 people so that one particular person gets 1 gallon and the rest of the syrup is divided equally among the remaining 4,
then each of those 4 people will get Y-1 / 4 gallons of syrup.
Ellen has received the following scores on 3 exams: 82, 74, and 90. What score will Ellen need to receive on the next exam so that the average (arithmetic mean) score for the 4 exams will be 85 ?
82+72+90+x / 4 = 85 =>( 246 + x)/ 4=85 x=94
A mixture of 12 ounces of vinegar and oil is 40 percent vinegar, where all of the measurements are by weight. How many ounces of oil must be added to the mixture to produce a new mixture that is only 25 percent vinegar?
(0.40 )( 12) / 12 + x = 0.25; x= 7.2
rectangular coordinate system
graph of an equation
Two lines are parallel if their slopes are equal. Two lines are perpendicular if their slopes are negative reciprocals of each other.
In the xy -plane above, the slope of the line passing through the points Q ( -2, - 3) and R ( 4, 1.5) is
1.5 - (-3) / 4 - (-2) = 0.75
parabola نمودار سهمی
( x - a )^2 + ( y - b )^2 = r^2
In general, for any function h ( x ) and any positive number c, the following are true.
• The graph of h ( x ) + c is the graph of h ( x ) shifted upward by c units. • The graph of h ( x ) - c is the graph of h ( x ) shifted downward by c units. • The graph of h ( x + c ) is the graph of h ( x ) shifted to the left by c units. • The graph of h ( x - c ) is the graph of h ( x ) shifted to the right by c units. • The graph of ch ( x) is the graph of h ( x) stretched vertically by a factor of c if c > 1. • The graph of ch ( x) is the graph of h ( x) shrunk vertically by a factor of c if 0 < c < 1.
Find an algebraic expression to represent each of the following. (a) The square of y is subtracted from 5, and the result is multiplied by 37. (b) Three times x is squared, and the result is divided by 7. (c) The product of ( x + 4) and y is added to 18.
a- What is the value of f ( x ) = 3x^2 - 7x + 23 when x = -2 ? (b) What is the value of h ( x ) = x^3 - 2 x^2 + x - 2 when x =2? (c) What is the value of k ( x) = 5/3 x - 7 when x = 0 ?
a- 49 b- 0 c- -7
Solve each of the following inequalities for x. (a) - 3 x > 7 + x (b) 25 x + 16 ≥ 10 - x (c) 16 + x > 8 x - 12
a- x < -7/4 b- x ≥ -3/13 c- x < 4
For a given two-digit positive integer, the tens digit is 5 more than the units digit. The sum of the digits is 11. Find the integer.
6 + 1 = 7 , 7 + 2 = 9 , 8 + 3= 11 so,83
If the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y ?
2x/5y = 3 / 4 => x/y = 15/8
Kathleen's weekly salary was increased by 8 percent to $237.60. What was her weekly salary before the increase?
237.60 / 1.08 = 220
Coordinates of the vertex
A theater sells children's tickets for half the adult ticket price. If 5 adult tickets and 8 children's tickets cost a total of $27, what is the cost of an adult ticket?
$3 5x+8y = 27 , y=1/2x => 5x+4x= 27=> x=3
In the xy-plane, find the following. (a) Slope and y-intercept of the line with equation 2 y + x = 6 (b) Equation of the line passing through the point (3, 2) with y-intercept 1 (c) The y-intercept of a line with slope 3 that passes through the point ( - 2, 1 ) (d) The x-intercepts of the graphs in (a), (b), and (c)
a- y=-1/2x+3 , so slope is -1/2 , y-intercept = 3 b- y = x/3 + 1 c- y-1=3(x+2) => y=6x+7 ; y-intercept = 7
For the parabola y = x^2 - 4x - 12 in the xy-plane, find the following. (a) The x-intercepts (b) The y-intercept (c) Coordinates of the vertex
a- x = 6 , x = -2 b- -12 c- x = (-b/2a , y ) = (2 , -16)
For the circle ( x - 1 )^2 + ( y + 1 )^2 = 20 in the xy-plane, find the following. (a) Coordinates of the center (b) Radius (c) Area
For the circle (x-a)^2 + (y-b)^2 = r^2, the center is (a, b), the radius is r and the area is pi*r^2. Therefore, for the circle (x-1)^2+ (y+1)^2 =20, The coordinates of the center are (1, -1) The radius is sqrt(20) The area is pi•(sqrt(20))^2 = 20 pi