 # Math 021.  An equation is defined as two algebraic expressions separated by an = sign.  The solution to an equation is a number that when substituted.

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Math 021

 An equation is defined as two algebraic expressions separated by an = sign.  The solution to an equation is a number that when substituted into the equation makes it a true statement.  For example, 8 is a solution to the equation x 2 – 9x = -8 since when x = 8 the equation becomes: 8 2 – 9(8) = -8 64 – 72 = -8 -8 = -8 which is a true statement However, 2 is not a solution since: 2 2 – 9(2) = -8 4 – 18 = -8 -14 = -8 which is a false statement

 A linear equation in one variable is any equation which contains a single variable and that variable is raised to the first power.  The general form of a linear equation in one variable is ax + b = c where a, b, and c are real numbers.

 Let a, b, c be real numbers. If a = b, then a + c = b + c. The addition property allows you to add or subtract any term from both sides of an equation and the equation will remain equal.  Examples – Solve the following using the addition property: a. x + 3 = 7 + 8b. 5x = 16 + 4x c. 7x - 5 = 8x + 10d. 10x – 5x = 4x – 11 e. 2(x + 6) = x – 3 f. 3(4x – 11) = -11(3 – x)

 Let a, b, c be real numbers, If a = b, then a∙c = b∙c. The multiplication property allows you to multiply or divide any non-zero number to both sides of an equation and the equation will remain equal.  Examples – Solve the following using the multiplication property: a. 7x = 35b. 5x + 6x = 39 + 5

c. -16(1-x) – 14x = –10 d. 5x – 4 = 26 + 2x e. 8x – 5x + 3 = x – 7 + 10 f. -2(5x – 1) – x = -4(x – 3)

 Multiply by a LCD to eliminate any fractions or multiply by a power of 10 to eliminate decimals  Use the distributive property if necessary  Combine like terms on the same side of the equal sign  Use the addition property to isolate the term containing the variable on one side of the equation and the real number to the other  Solve for the variable by using the multiplication property

 Examples – Solve each of the following: a. b. c. d. =

 e. 0.5x – 0.3 = 1.1 + 0.3x  f. 0.15(4 – x) = 0.13(2 – x)

 A contradiction is a statement in mathematics that when completely simplified is false. A linear equation that simplifies to a contradiction has no solution.  An identity is a statement in mathematics that when completely simplified is always true. A linear equation that simplifies to an identity has an infinite number of solutions, or all real numbers.

 a. 5x – 6x – 3 = -(x + 3)  b. 3x + 3 + 5 = 2x + 2 + x  c. 9(x – 2) = 7(x – 10) + 2x  d. 5(x – 4) + x = 6(x – 2) – 8

 An absolute value equation is any equation that contains one or more absolute values.  To eliminate absolute values, use the definition that if |x| = c, then x = c or x = -c  Examples – Solve each of the following:  a. |x + 3| = 7b. |x + 3| – 4 = 7  c. |2x – 5| + 1 = 6d. 3|5 – x| – 1 = 8  e. |3x + 1| + 10 = 6f. |2x| = -15

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