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Systems of liner equations: The equations X+Y =10 & X-Y= 2, each have lost of solutions (infinitely may, in fact)
However, there is only one pair of numbers, X=6 & Y=4, that satisfy both equations at the same time. Hence this is the only solution of the system of equations
X+Y=10
X-Y=2
A system of equations is a set of two or more equations involving two or more variables. Solving such a system means finding a value for each variable that will make each equation true.
One common way of solving a system of equations is to replace a variable by its expression in the other equation. Let us again take the system.
X+Y=10 .........................(1)
X-Y=2 ............................(2)
From (2) X=2+Y
Replacing this value of X in equation (1)
2+Y+Y =10
Or, 2Y = 10-2 =8
:. Y=8/2=4
:. X = 2+Y=2+4= 6
:. The solution of this system of equations is (X, Y) = (6, 4)
Quadratic Equations:
When the highest power of the variable in an equation is 2 (i.e square) it is called a quadratic equation. e.g.
(9x2)+ 2.(2x). 1+12 = 9
Or, (2X+1)2= 9
Or, 2X+1 = 3
Or, 2X= 3-1 =2, -4
:. X=1 or-2
The equation 3X2 +5X + 2 = 0 cannot be solved in the above mentioned way. We will use middle term factoring to solve this.
3X2 + 5X + 2 = 0
Or, 3X2 +3X + 2X + 2 =0
Or, 3X(X+1) + 2(X+1)= 0
Or, (3X+2)(X+1) = 0
Or, 3X+2 = 0 Or, X+1 =0
Or, X =-(2/3) Or, X = -1
Answer: X= -(2/3) or -1
Inequalities:
We denote the statement is greater than B by a > b. this means that in the number line,a is to the right of B. Similarly, a is less than b is denote a < b, If a is to the left of b on the number line. So if a is positive, we can write a > 0, If a is negative, a < 0.
For any number a, b:
a > b means that a - b > 0
and a < b means that a - b < 0
For any numbers a and b, exactly one of the following is true: a > b, a = b or a < b.
We have another pair of inequality symbols:
≤ (Grater than or equal to )
≥ ( Smaller than or equal to )
So, X ≤ 5 means, X can take any value that is smaller than 5 or it can be 5 itself. Sometimes we may combine inequality statements. For example, 2 < X and X< 5 can be combined to write 2< X<5. This means that X is a number between 2 and 5.
Manipulating inequalities are a bit complex compared to equalities or equations. An arithmetic operation on an inequality may preserve or reverse it. If the result of performing an arithmetic operation on an inequality results in a new inequality in the same direction, the inequality has been preserved. On the other hand if the arithmetic operation reverses the direction of the inequality, then we say that the inequality has been reversed. Now we will see a few basic facts about preserving & reversing inequalities.
<> Adding / Subtracting a number to an inequality preserves it.
:. If a < b, then a+ c < b + c and a- c < b -c
e.g. 3 < 7 => 3+ 100 < 7 + 100 and 3 - 100 < 7 -100
<> Adding inequalities in the same direction preserves it.
If a < b and c < d, then a + c < b + d
e.g If 5 < 10 and 7 < 8 => 5 + 7 < 10 +8
Note that this rule is not true for subtraction.
9 < 10 and 7 < 8.5
But (9 - 7) > ( 10 - 8.5)
<> Multiplying or dividing an equality bya positive number preserves it
If a > b & c is positive, then ac > bc and a/c > b/c
:. 9 > 5 => 9 x 4 > 5 x 4 and 9/4 > 5/4
<> Multiplying or dividing an inequality by a negative umber reverses it
If a < b and c is negative then ac > bc and a/c > b/c
e.g. 5 < 9 => 5 x - 4 > 9 x - 4 and 5/-4 > 9/-4
<> Taking negative reverses and inequality.
If a > b, then -a < -b
e.g. 9 > 6 => - 9 < -6
This is actually an extension of the previous fact. Taking negative is nothing but multiplication by (-1).
<>If two numbers are each positive or negative, then taking reciprocals reverses an inequality.
If a and b are either positive or negative and a > b, then 1/a < 1/b
e.g 5 > 2 => 1/5 < 1/2
-5 < - 2 => -1/5 > -1/2
The numbers between 0 & 1 has special properties.
<> If 0 < X < 1 and a is positive, then xa < a.e.g.0.58 x 19 x < 19
<> If 0 < X < 1 and m & n are integers with m > n > 1 , then Xm < Xn < X
e.g. (0.5)3 < (0.5)2 < (0.5)1
<> If 0 < X < 1, then 1/x > 1
e.g 1/0.25 > 0.25 & 1/0.25 > 1
X+Y=10
X-Y=2
A system of equations is a set of two or more equations involving two or more variables. Solving such a system means finding a value for each variable that will make each equation true.
One common way of solving a system of equations is to replace a variable by its expression in the other equation. Let us again take the system.
X+Y=10 .........................(1)
X-Y=2 ............................(2)
From (2) X=2+Y
Replacing this value of X in equation (1)
2+Y+Y =10
Or, 2Y = 10-2 =8
:. Y=8/2=4
:. X = 2+Y=2+4= 6
:. The solution of this system of equations is (X, Y) = (6, 4)
Quadratic Equations:
When the highest power of the variable in an equation is 2 (i.e square) it is called a quadratic equation. e.g.
(9x2)+ 2.(2x). 1+12 = 9
Or, (2X+1)2= 9
Or, 2X+1 = 3
Or, 2X= 3-1 =2, -4
:. X=1 or-2
The equation 3X2 +5X + 2 = 0 cannot be solved in the above mentioned way. We will use middle term factoring to solve this.
3X2 + 5X + 2 = 0
Or, 3X2 +3X + 2X + 2 =0
Or, 3X(X+1) + 2(X+1)= 0
Or, (3X+2)(X+1) = 0
Or, 3X+2 = 0 Or, X+1 =0
Or, X =-(2/3) Or, X = -1
Answer: X= -(2/3) or -1
Inequalities:
We denote the statement is greater than B by a > b. this means that in the number line,a is to the right of B. Similarly, a is less than b is denote a < b, If a is to the left of b on the number line. So if a is positive, we can write a > 0, If a is negative, a < 0.
For any number a, b:
a > b means that a - b > 0
and a < b means that a - b < 0
For any numbers a and b, exactly one of the following is true: a > b, a = b or a < b.
We have another pair of inequality symbols:
≤ (Grater than or equal to )
≥ ( Smaller than or equal to )
So, X ≤ 5 means, X can take any value that is smaller than 5 or it can be 5 itself. Sometimes we may combine inequality statements. For example, 2 < X and X< 5 can be combined to write 2< X<5. This means that X is a number between 2 and 5.
Manipulating inequalities are a bit complex compared to equalities or equations. An arithmetic operation on an inequality may preserve or reverse it. If the result of performing an arithmetic operation on an inequality results in a new inequality in the same direction, the inequality has been preserved. On the other hand if the arithmetic operation reverses the direction of the inequality, then we say that the inequality has been reversed. Now we will see a few basic facts about preserving & reversing inequalities.
<> Adding / Subtracting a number to an inequality preserves it.
:. If a < b, then a+ c < b + c and a- c < b -c
e.g. 3 < 7 => 3+ 100 < 7 + 100 and 3 - 100 < 7 -100
<> Adding inequalities in the same direction preserves it.
If a < b and c < d, then a + c < b + d
e.g If 5 < 10 and 7 < 8 => 5 + 7 < 10 +8
Note that this rule is not true for subtraction.
9 < 10 and 7 < 8.5
But (9 - 7) > ( 10 - 8.5)
<> Multiplying or dividing an equality bya positive number preserves it
If a > b & c is positive, then ac > bc and a/c > b/c
:. 9 > 5 => 9 x 4 > 5 x 4 and 9/4 > 5/4
<> Multiplying or dividing an inequality by a negative umber reverses it
If a < b and c is negative then ac > bc and a/c > b/c
e.g. 5 < 9 => 5 x - 4 > 9 x - 4 and 5/-4 > 9/-4
<> Taking negative reverses and inequality.
If a > b, then -a < -b
e.g. 9 > 6 => - 9 < -6
This is actually an extension of the previous fact. Taking negative is nothing but multiplication by (-1).
<>If two numbers are each positive or negative, then taking reciprocals reverses an inequality.
If a and b are either positive or negative and a > b, then 1/a < 1/b
e.g 5 > 2 => 1/5 < 1/2
-5 < - 2 => -1/5 > -1/2
The numbers between 0 & 1 has special properties.
<> If 0 < X < 1 and a is positive, then xa < a.e.g.0.58 x 19 x < 19
<> If 0 < X < 1 and m & n are integers with m > n > 1 , then Xm < Xn < X
e.g. (0.5)3 < (0.5)2 < (0.5)1
<> If 0 < X < 1, then 1/x > 1
e.g 1/0.25 > 0.25 & 1/0.25 > 1
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