Math 017 Materials With Exercises

Operations that can be performed on any equation without changing the set of solutionsThe following operations can be done to any equation without changing its solution.Any quantity can be added to, or subtracted from both sides of an equation.For example, If x y then x 2 y 2If x y then x 2 y 2Both sides of an equation can be multiplied, or divided by any nonzero quantity.For example, If x y then 2x 2yx yIf x y then 2 2It is important to remember that, any time we perform any operation on one side of an equation,exactly the same operation must be performed on the other side. We always perform the sameoperation on both sides.Solving linear equations in one variableThere are many types of equations, and depending on the type of equation, different solving techniquesare involved. We will learn how to solve linear equations in one variable (the formal definition of alinear equation will be provided later).To solve a linear equation we perform the operations of adding, subtracting, multiplyingor dividing both sides of an equation by suitable quantities with the goal of isolating thevariable (often x) on one side, and „bringing all numbers to the other side‟ of the equation.The numerical value obtained on the other side is the solution.We will learn how to implement the above idea by solving several equations.■ Solve the following equation x 2 7 .We need to isolate x , and to this end, we must somehow „get rid of 2‟ on the left-hand side of theequation. 2 is added to x , so if we subtract 2 (opposite operation to addition), we will get zero, and 2will no longer be on the left-hand side. But if we subtract 2 from the left-hand side of the equation, thesame operation must be performed on the right-side. Thus we need to subtract 2 from both sides. Wecould say that we are „bringing 2 to the other side‟x 2 7Subtract 2 from both sides.x 2 2 7 2Simplify.x 5The solution of x 2 7 is x 5.■79