Solving for the Unknown: A How-to Approach for Solving Equations Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

Learning Unit Objectives #5 Solving for the Unknown: A how-to Approach for Solving Equations Learning Unit Objectives LU5.1 Solving Equations for the Unknown Explain the basic procedures used to solve equations for the unknown List the five rules and the mechanical steps used to solve for the unknown in seven situations; know how to check the answers

Learning Unit Objectives #5 Solving for the Unknown: A how-to Approach for Solving Equations Learning Unit Objectives LU5.2 Solving Word Problems for the Unknown List the steps for solving word problems Complete blueprint aids to solve word problems; check the solutions

Terminology Expression – A meaningful combination of numbers and letters called terms. Equation – A mathematical statement with an equal sign showing that a mathematical expression on the left equals the mathematical expression on the right. Formula – An equation that expresses in symbols a general fact, rule, or principle. Variables and constants are terms of mathematical expressions.

Solving Equations for the Unknown Equality in equations A + 8 58 Left side of equation Right side of equation Dick’s age in 8 years will equal 58

Variables and Constants Rules 1. If no number is in front of a letter, it is a 1: B = 1B; C = 1C 2. If no sign is in front of a letter or number, it is a +: C = +C; 4 = +4

Solving for the Unknown Rule Whatever you do to one side of an equation, you must do to the other side.

Opposite Process Rule If an equation indicates a process such as addition, subtraction, multiplication, or division, solve for the unknown or variable by using the opposite process.

Opposite Process Rule A + 8 = 58 - 8 - 8 A = 50 Check 50 + 8 = 58

Equation Equality Rule You can add the same quantity or number to both sides of the equation and subtract the same quantity or number from both sides of the equation without affecting the equality of the equation. You can also divide or multiply both sides of the equation by the same quantity or number (except 0) without affecting the equality of the equation.

Equation Equality Rule 7G = 35 7 7 G = 5 Check 7(5) = 35

Multiple Processes Rule When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division.

( ) Multiple Process Rule H + 2 = 5 4 -2 -2 H = 3 H = 4(3) H = 12 (4) -2 -2 H = 3 H = 4(3) H = 12 ( ) (4) Check 12 + 2 = 5 4 3 + 2= 5

Parentheses Rule When equations contain parentheses (which indicates grouping together, you solve for the unknown by first multiplying each item inside the parentheses by the number or letter just outside the parentheses. Then you continue to solve for the unknown with the opposite process used in the equation. Do the addition and subtractions first; then the multiplication and division.

Parentheses Rule 5(P - 4) = 20 5P – 20 = 20 +20 +20 5P = 40 5 5 P =8 +20 +20 5P = 40 5 5 P =8 Check 5(8-4) = 20 5(4) = 20 20 = 20

Like Unknown Rule To solve equations with like unknowns, you first combine the unknowns and then solve with the opposite process used in the equation.

Like Unknown Rule 4A + A = 20 5A = 20 5 5 A = 4 Check 4(4) +4 = 20 5 5 A = 4 Check 4(4) +4 = 20 16 + 4 = 20

Solving Word Problems for Unknowns 1) Read the entire Problem 3) Let a variable represent the unknown Y = Computers 4) Visualize the relationship between the unknowns and variables. Then set up an equation to solve for unknown(s) Read again if necessary 2) Ask: “What is the problem looking for?” 4Y + Y = 600 5) Check your results to ensure accuracy

Solving Word Problems for the Unknown Blueprint aid

Solving Word Problems for the Unknown ICM Company sold 4 times as many computers as Ring Company. The difference in their sales is 27. How many computers of each company were sold? Cars Sold ICM 4C 4C Ring C -C 27 Ring = 9 computers ICM = 4(9) = 36 Computers 4C - C = 27 3C = 27 3 3 C = 9 Check 36 - 9 = 27

Unknown(s) Variable(s) Relationship Problem 5-34: Solution: 4S + S = 5,500 = 5,500 S = 1,100 4S = 4,400 5S 5 Unknown(s) Variable(s) Relationship Shift 1 4S 4S (4,400) Shift 2 S + S (1,100) 5,500

Unknown(s) Variable(s) Relationship Problem 5-36: Solution: T + 3T = $40,000 = $40,000 4 T = $10,000 3T = $30,000 4T Unknown(s) Variable(s) Relationship Jim T T ($10,000) Phyllis 3T + 3T ($30,000) $40,000

Unknown(s) Variable(s) Relationship Problem 5-37: Solution: 1.5L + L = 5,600 = L = 2,240 1.5L = 3,360 2.5L 2.5 5,600 Unknown(s) Variable(s) Relationship Shift 1 1.5L 1.5L (3,360) Shift 2 L +L (2,240) 5,600

Unknown(s) Variable(s) Price Relationship Problem 5-38: Solution: Check: 60($6) + 420($2) = $1,200 $360 + $840 = $1,200 $1,200 = $1,200 14B + 6B = 1,200 = B = 60 bottles 7B = 420 thermometers 20B 20 1,200 Unknown(s) Variable(s) Price Relationship Thermometers 7B $2 14B Hot-water Bottles B 6 +6B Total = $1,200