Introduction to Equations

Contemporary Math

• Choose any 2 numbers for the left part of the scale and 2 numbers for the right part of the scale so that the 2 sums are equal.

• 1. 22, 29, 46, 53• 2. 32, 34, 53, 55• 3. 22, 38, 61, 64, 77• 4. 86, 91, 98, 102, 103

• In Math, equation – statement that the 2 expressions are equal.

• In the expression

x + 3 = 5

Left-hand side

Right-hand side

To solve an equation means to find all the solutions to the equation.

Definition

Equivalent equations – are equations that have the same solution.

Ex. x + x = 2x , 3 = 3

Inconsistent equation – equation with no solutionconsistent equation – equation that has at least

one solution that is not an identity

Root- any value of the variable that makes the equation a true statement.

Ex. 2x – 4 = 0; x / 3 + 5 = 6

Examples

• Is the given number a solution to the equation?

1) x + 5 = 10 ; 55 + 5 =

1010= 10

Yes.

2) n – 5 = 12 ; 1818- 5 = 1213 12

No.

3) 5x – 11 =29; 84) x / 9 = 6 ; 3

• Is the given number a solution to the equation?

• x + 5 = 10; 55+ 5 = 1010= 10yes

• n – 5 = 12; 1818 – 5 = 12

13 12no

• 5x – 11 = 29 ; 85(8) -11 = 29

40 – 11 = 2929 = 29yes

• x/9 = 6; 6363/ 9 = 67 6no

• Tell whether the equation is inconsistent equation, conditional equation, or an identity.

• 1. x + 1 = x + 1 Identity• 2. x = x + 1 Inconsistent. If we replace x

by 4, we get 4 = 4 + 1 which is false.• 3. x – 3 = 8 conditional. The

statement x – 3 = 8 is true only on condition that x = 11.

• 4. 2x + 3 = 11 Conditional. The statement 2x + 3 =11 is true only on condition that x = 4.

+

-

x

Increased by, sum, plus, added to , more than, greater than

Difference, subtract, minus, less, diminished, decreased, take away, reduced

Multiply, product of

Divide, quotient

• English sentence to mathematical sentence.

• The sum of m and 9 m + 9• 10 added to m m + 10• 7 plus x 7 + x• 12 more than n n + 12• 9 increased by z 9 + z• 25 greater than n n + 25• Exceeds z by 100 z + 100• The difference of 2 and x 2 – x• 20 subtracted from c c – 20• 8 minus f 8 – f• 3 less than c c – 3• 5 decreased by p 5 – p• 53 take away k 53 - k

English sentence to mathematical sentence.

• Write a mathematical equation for each sentence.

• A number increased by 7 is 25. X + 7 = 25Eleven diminished by a number is 53 11- n = 53Five times a number is 45.5x = 45The quotient of a number and 8 is 12n/ 8 = 12

• Is the number in the parenthesis a solution to the given equation

• 1. a + 9 = 12 (21)• 2. 8 + b = 30 (22)• 3. -7c = -3 (4)• 4) 3d + 15 = 3 (-4)• 5) 4e – 8 = 3e – 4 (4)• 6) 12 f = -12 (- 1)• 7) 11g = -77 (7)• 8) 5x – 4 = 9 (7)• 9) 8j – 2 = -10 (-1)• 10) 5x – 4 = 9 (7)