![]() Not all equations can be solved mentally. Not so! The final step should always be to check the solution. ![]() Many students think that when they have found the solution to an equation, the problem is finished. This is a true statement, so x = 5 is correct. This is not a true statement, so the answer x = 6 is wrong.Īnother student solved the same equation and found x = 5. Was this right or wrong?ĭoes x = 6 satisfy the equation 5x - 3 = 4x + 2? To check we substitute 6 for x in the equation to see if we obtain a true statement. Regardless of how an equation is solved, the solution should always be checked for correctness.Įxample 11 A student solved the equation 5x - 3 = 4x + 2 and found an answer of x = 6. What number do we divide 2 by to obtain 7? Our answer is 14. What number must be multiplied by 3 to obtain 15? Our answer is x = 5. What number do we subtract 5 from to obtain 3? Again our experience with arithmetic tells us that 8 - 5 = 3. Therefore the solution to the equation is x = 4. Our knowledge of arithmetic indicates that 4 is the needed value. To have a true statement we need a value for x that, when added to 3, will yield 7. The better you know the facts of multiplication and addition, the more adept you will be at mentally solving equations. Ability to solve an equation mentally will depend on the ability to manipulate the numbers of arithmetic. Solving an equation means finding the solution or root. The solution or root is said to satisfy the equation. The literal numbers in an equation are sometimes referred to as variables.įinding the values that make a conditional equation true is one of the main objectives of this text.Ī solution or root of an equation is the value of the variable or variables that make the equation a true statement. ![]() A conditional equation is true for only certain values of the literal numbers in it.Įxample 4 x + 3 = 9 is true only if the literal number x = 6.Įxample 5 3x - 4 = 11 is true only if x = 5. An identity is true for all values of the literal and arithmetical numbers in it.Įxample 3 2x + 3x = 5x is an identity since any value substituted for x will yield an equality.Ģ. Determine if certain equations are equivalent.Īn equation is a statement in symbols that two number expressions are equal.Įquations can be classified in two main types:ġ.Classify an equation as conditional or an identity.Upon completing this section you should be able to: CONDITIONAL AND EQUIVALENT EQUATIONS OBJECTIVES To accomplish this we will use the skills learned while manipulating the numbers and symbols of algebra as well as the operations on whole numbers, decimals, and fractions that you learned in arithmetics. In this chapter we will study some techniques for solving equations having one variable. The solution of equations is the central theme of algebra.
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