Linear equations and inequalities
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Linear equations and inequalities are mathematical expressions that involve variables raised to the power of 1. They represent relationships between different quantities and are widely used in various fields of study, including mathematics, physics, economics, and engineering.
A linear equation is an equation in which the variables with an exponent of 1 are raised to the power of 1. It can be represented in the form: ax + b = 0, where "a" and "b" are constants, and "x" is the variable. For example, 2x + 3 = 9 is a linear equation.
Solving a linear equation involves finding the value of the variable that satisfies the equation. One common method is to isolate the variable by performing operations such as addition, subtraction, multiplication, and division on both sides of the equation. For example, to solve 2x + 3 = 9, we can subtract 3 from both sides to get 2x = 6, and then divide both sides by 2 to find x = 3.
On the other hand, a linear inequality is an inequality involving variables raised to the power of 1. It can be represented in the form: ax + b < c, ax + b > c, ax + b ≤ c, or ax + b ≥ c, where "a," "b," and "c" are constants, and "x" is the variable. For example, 2x + 3 < 9 is a linear inequality.
To solve a linear inequality, the process is similar to solving a linear equation. However, there is a difference when multiplying or dividing by a negative number. If you multiply or divide both sides of the inequality by a negative number, the direction of the inequality symbol should be reversed. For example, if we divide both sides of 2x + 3 < 9 by -2 (a negative number), we get x > -3.
In both linear equations and inequalities, there are different techniques and methods you can use to solve them, such as substitution, elimination, graphing, or using matrices. The choice of method often depends on the complexity of the equation or inequality.