Solving Equations in One Variable Unit 1.2 Solving Equations in One Variable

FOUNDATIONS OF EQUATIONS An equation is a mathematical statement that two expressions are the same. Inverse operations: Addition/Subtraction Multiplication/Division Squaring/Square root Equations are balanced – whatever is done to one side MUST be done to the other side.

FOUNDATIONS cont. To solve an equation, isolate the variable by using inverse operations. REGULAR EQUATIONS: One step: only one operation to undo Example: x + 6 = 10 2. y – 5 = 15 -6 -6 + 5 + 5 x = 4 y = 20 -

FOUNDATIONS cont. Example: 2x + 5 = 17 Two step equations: Undo the add/subtract and then undo the multiply/divide Example: 2x + 5 = 17 - 5 - 5 2x = 12 ÷ 2 ÷ 2 x = 6

SOME TRICKY EQUATIONS… Negative variable equations: -x = 35 A number minus variable equations: -3 – x = 32

MORE TRICKY EQUATIONS… Fraction coefficient equations:

MORE TRICKY EQUATIONS… Big fraction bar:

COMBINING LIKE TERMS IN EQUATIONS… Group and combine like terms on both sides of the equation (separately) before solving. Remember: When working on the same side of the equal sign, there is no need for inverse operations! Example: 2x + 4 + 5x – 10 = 15

COMBINING LIKE TERMS IN EQUATIONS… 6 – 2x + 10 = 20 + 8 -3w – 8 + 5w = -14

DISTRIBUTIVE PROPERTY IN SOLVING EQUATIONS… Clear the ()s first by using the distributive propery, Then solve for the variable Example: 4(3x + 1) = 40

DISTRIBUTIVE PROPERTY IN SOLVING EQUATIONS… 6x – 5(2x – 2) = 38 4(3x + 5) = 40

SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES… The goal is to get the variables on the same side of the equation, and then solve as any other multi-step equation. The variables can be added/subtracted like integers.

SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES… 5x + 2 = 2x + 14 7k + 2 = 4k – 10

SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES… 2(5x – 1) = 3(x + 11)

SOLVING EQUATIONS INVOLVING FRACTIONS… To clear the fractions in an equation: Find the LCM of all of the denominators. Multiply each term by the LCM of the denominators. (don’t forget to multiply terms that are not written as fractions too) Solve your equation.

SOLVING EQUATIONS INVOLVING FRACTIONS…

SOLVING EQUATIONS INVOLVING FRACTIONS…

PUTTING IT ALL TOGETHER… Plastic Fruit Looks Very Real This is the order to do things when your multistep equation is very complicated… Parentheses Fractions Like Terms Variable on one side Regular 1 or 2 step

PUTTING IT ALL TOGETHER…

INFINITE OR NO SOLUTION… 10x + 12 = 2(5x + 6) 9m – 4 = -3m + 5 + 12m

ABSOLUTE VALUE EQUATIONS… Absolute value is the distance that a number is from zero on the number line. Absolute value does not care about direction and, therefore, there is not a negative distance. It is a measure of units. NO NEGATIVES!!!

EXAMPLES OF ABSOLUTE VALUE… | 7 | | - 7| | -X|

SOLVING ABSOLUTE VALUE EQUATIONS… Clear the equation with the | | so that only the | | and its contents are on that side. Let the contents of the | | = the positive and negative value of the number on the opposite side of the equation. Solve both equations

SOLVING ABSOLUTE VALUE EQUATIONS…