Algebra: Linear equations and inequalities

Welcome back to my blog!

Today, we're continuing our journey into the world of algebra. Many problems in math can be solved using equations. We convert word problems into algebraic equations by representing unknown quantities with variables such as "x." We then follow a formal procedure to find the solution. A linear equation typically takes the form ax+b=0, where x is a variable "a" and "b" are constants, and a≠0. Examples of linear equations include 3x-2=4 and 2x+1=9.

Note: There is a crucial difference between equations and expressions. Expressions are representations of mathematical statements that are not equated to anything, while equations are expressions equated to something. In other words, there's an equal to(=) sign in equations but there's no equal to(=) sign in expressions.

Sides of an Equation

Every equation has two sides: the left-hand side (LHS) and the right-hand side (RHS). For example, in the equation 3x+4=9

LHS: 3x+4

RHS: 9

Solution of an equation

The solution of an equation is the value of the variable that makes the equation true, meaning it makes the LHS equal to the RHS.

Example : For the equation : 2x+5=9, the value of x that satisfies this is 2:

=》2(2) + 5 = 4 + 5 = 9. Therefore, 2 is the solution to this equation.

Example 1

Solve the following equations:

i. 3x +7=x+13

Grouping like terms:

3x - x = 13 - 7

=》2x = 6

=》2x/2= 6/2

ii. 5x+10 = 2x+13

Rearranging gives:

5x - 2x = 13 - 10

=》3x = 3

=》3x/3 =3/3

=》x=1

iii. (x/3) +2= -2

=》x/3 = -2-2

=》x/3 = -4

=》3×(x/3) = -4×3

=》x = -12

iv. 5(x+1) -2x = -7

=》5x+5 -2x = -7

=》 3x = -7-5

=》3x = -12

=》3x/3 = -12/3

=》x = -4

Word Problems

To solve word problems, convert the statements into equations. Here are a few examples:

i.When a number is added to 5, the result is 20.

Let the number be y. Next,we add 5 and equate it to 20

=》y + 5 = 20

ii. Twice a certain number is 5 more than the number.

Let the number be z. Twice of z is equal 5 more than z

=》2z = z + 5

iii. The sum of two consecutive even numbers is 30.

                                Solution

Let the first even number be x and the second number be (x+2).

Proof;

Consider a set of even numbers, E={2,4,6,8}. The difference between an even number and the preceding one is 2.

i.e ; 4-2 =2, 6-4=2.

Therefore, to find the subsequent term in a sequence of even numbers, we need to add 2 to the current term.

=》x + (x + 2) = 30

=》2x + 2 = 30

iv. A table costs 13 cents each and a chair costs 11 cents each. If I buy 5 more tables than chairs, the total cost is $2.50

                               Solution

Let t be the number of tables and c be the number of chairs:

=》13t + 11c = $2.50

We need to convert $2.50 into cents since the prices of the table and chair were given in cents

=》$2.50 = 250 cents and also, t = c + 5

=》13(c+5) +11c =250

Linear Inequalities

Linear inequalities involve the following symbols:less than or equal to(≤), less than(<), greater than(>), greater than or equal to(≥)

Important Rule : If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Example

Find the solutions of the following inequalities.

i. 3x -4 ≥ 2

=》3x ≥ 2+4

=》3x ≥ 6

=》3x/3 ≥ 6/3

=》 x ≥ 2

ii. 3 - 2x < 7

=》-2x < 7-3

=》-2x < 4

=》(-2x/-2) < (4/-2)

=》x > -2

iii. 5 - 3x ≤ 2x+7

=》-3x -2x ≤ 7-5

=》 -5x ≤ 2

=》-5x/-5 ≤ 2/-5

=》x ≥ -2/5

iv. -5 < 9-2y

=》2y < 9 +5

=》2y < 14

=》2y/2 < 14/2

=》y < 7

                              Assignment

1. When 7 times a number is increased by 11 and the result is 31 more than the number, find the number.

2. The sum of three consecutive integers is 63. Find the largest integer.

3. Solve for y in the following

i. (2y+1)/3 -(4-x)/6 = -2

ii. 7 - 2(y-3) > 5(3-2y)

iii. 2(y -2) -5(y+3) = -5

iv. 22y +7 ≤ 22-6y

4. Clara, Dean, and Elaine were candidates in an election in which 1000 people voted. Elaine won, receiving 95 more votes than Dean and 186 more than Clara. How many votes did Dean receive?

Algebra is a powerful tool that helps us understand and solve various problems. Practice these concepts and examples to strengthen your skills and don't hesitate to explore further. You can also reach out to ask questions about anything you don’t understand under this topic for further clarification. Happy learning.