Chapter 3Chapter 3Solving LinearSolving Linear

EquationsEquations

3.1 Solving Equations Using 3.1 Solving Equations Using Addition and SubtractionAddition and Subtraction

You can solve an equation by using You can solve an equation by using the the transformationstransformations below to isolate below to isolate the variable on one side of the the variable on one side of the equation.equation.

When you rewrite an equation using When you rewrite an equation using these transformations, you produce these transformations, you produce an equation with the same solutions an equation with the same solutions as the original equation.as the original equation.

These equations are called These equations are called equivalentequivalent equations. equations.

To change, or To change, or transformtransform, an equation , an equation into an equivalent equation, think of into an equivalent equation, think of an equation as having two sides that an equation as having two sides that are “in balance.”are “in balance.”

Any transformation you apply to an Any transformation you apply to an equation must keep the equation in equation must keep the equation in balance.balance.

For example, if you if you subtract 3 For example, if you if you subtract 3 from one side of the equation, you from one side of the equation, you must also subtract 3 from the other must also subtract 3 from the other side of the equation.side of the equation.

Transformations that produce Transformations that produce Equivalent EquationsEquivalent Equations

Original Original EquationEquation

Equivalent Equivalent EquationEquation

Add the Add the same # to same # to each sideeach side

xx – 3 = 5 – 3 = 5

Subtract Subtract the same # the same # to each sideto each side

xx + 6 = 10 + 6 = 10

Simplify Simplify one or both one or both sidessides

x x = 8 – 3 = 8 – 3

Interchange Interchange the sidesthe sides

7 = 7 = xx

Example 1Example 1: : Solve. Solve. a) a) xx – 5 = -13 – 5 = -13 b) b) xx – 9 = -17 – 9 = -17

Example 2Example 2: : Solve.Solve. a) -8 = a) -8 = nn – (-4) – (-4)

b) -11 = b) -11 = nn – (-2) – (-2)

Linear Equations: Linear Equations: The variable is raised The variable is raised to the to the firstfirst power and does not occur in a power and does not occur in a denominator, inside a square root, or denominator, inside a square root, or inside absolute value symbols.inside absolute value symbols.

Linear EquationLinear Equation Not a Linear Not a Linear EquationEquation

x x + 5 = 9+ 5 = 9 xx22 + 5 = 9 + 5 = 9

- 4 + - 4 + nn = 2 = 2nn – 6 – 6 ||xx + 3| = 7 + 3| = 7

Example 3Example 3: : Several record Several record temperature changes have taken temperature changes have taken place in Spearfish, South Dakota. On place in Spearfish, South Dakota. On January 22, 1943, the temperature in January 22, 1943, the temperature in Spearfish fell from 54Spearfish fell from 54°F at 9:00am to °F at 9:00am to -4°F at 9:27am. By how many -4°F at 9:27am. By how many degrees did the temperature fall? degrees did the temperature fall?

Example 4Example 4: : Match the real-life problem Match the real-life problem with an equation.with an equation.

xx – 4 = 16 – 4 = 16 xx + 16 = 4 16 – + 16 = 4 16 – xx = 4 = 4 a) You owe $16 to your cousin. You paid a) You owe $16 to your cousin. You paid xx

dollars back and now you owe $4. How dollars back and now you owe $4. How much did you pay back?much did you pay back?

b) The temperature was b) The temperature was xx°F. It rose 16°F °F. It rose 16°F and is now 4°F. What was the original and is now 4°F. What was the original temperature?temperature?

c) A telephone pole extends 4 feet below c) A telephone pole extends 4 feet below ground and 16 feet above ground. What is ground and 16 feet above ground. What is the total length the total length xx of the pole? of the pole?

3.2 Solving Equations using 3.2 Solving Equations using Multiplication and DivisionMultiplication and Division

Original Original EquationEquation

EquivalenEquivalent Equationt Equation

Multiply Multiply each each equation equation by the by the same same nonzero #nonzero #

Divide Divide each each equation equation by the by the same same nonzero #nonzero #

44xx = 12 = 12

32

x

Example 1Example 1: : Solve.Solve. a) - 4a) - 4xx = 1 = 1 b) 7b) 7nn = - 35 = - 35

Example 2Example 2: : Solve.Solve. a) a)

b) b)

305

x

37

f

Example 3Example 3: : Solve.Solve.

a) a)

b) b)

94

3 t

m3

210

Properties of EqualityProperties of Equality:: Addition Property:Addition Property:

Subtraction Property:Subtraction Property:

Multiplication Property:Multiplication Property:

Division Property:Division Property:

3.3 Solving Multi-Step Equations3.3 Solving Multi-Step Equations

Solving a linear equation may require Solving a linear equation may require two or more transformations.two or more transformations.

Simplify one or both sides of the Simplify one or both sides of the equation (if needed).equation (if needed).

Use the inverse operations to isolate Use the inverse operations to isolate the variable.the variable.

Example 1Example 1: : Solve.Solve. a) a)

b) b)

Example 2Example 2: : Solve.Solve. a) 7a) 7xx – 3 – 3xx – 8 = – 8 =

2424

b) 2b) 2xx – 9 – 9xx + 17 = - + 17 = - 44

863

1x

1052

1x

Example 3Example 3: : Solve.Solve. a) 5a) 5xx + 3( + 3(xx + 4) = 28 + 4) = 28

b) 4b) 4xx – 3( – 3(xx – 2) = 21 – 2) = 21

Example 4Example 4: : Solve.Solve. a) 4a) 4xx + 12( + 12(xx – 3) = 28 – 3) = 28

b) 2b) 2xx – 5( – 5(xx – 9) = 27 – 9) = 27

Example 5Example 5: : Solve.Solve. a) a)

b) b)

Example 6Example 6: : A body temperature of 95A body temperature of 95°F °F or lower may indicate the medical or lower may indicate the medical condition called hypothermia. What condition called hypothermia. What temperature in the Celsius scale may temperature in the Celsius scale may indicate hypothermia?indicate hypothermia?

)3(5

666 x

)2(10

312 x

Example 7Example 7: : The temperature within The temperature within Earth’s crust increases about 30Earth’s crust increases about 30° ° Celsius for each kilometer of depth Celsius for each kilometer of depth beneath the surface. If the beneath the surface. If the temperature at Earth’s surface is temperature at Earth’s surface is 24°C, at what depth would you 24°C, at what depth would you expect the temperature to be 114°C?expect the temperature to be 114°C?

3.6 3.6

Objective: To find exact and Objective: To find exact and apporoximate solutions of equations apporoximate solutions of equations that contain decimals.that contain decimals.

Round-off ErrorRound-off Error: :

Example 1Example 1: : Solve the equation. Solve the equation. Round to the nearest hundredth.Round to the nearest hundredth.

a) 7.23x + 16.51 = 47.89 – 2.55xa) 7.23x + 16.51 = 47.89 – 2.55x

Example 1Example 1: : Solve the equation. Solve the equation. Round to the nearest hundredth.Round to the nearest hundredth.

a) 7.23x + 16.51 = 47.89 – 2.55xa) 7.23x + 16.51 = 47.89 – 2.55x

b) 6.6(1.2 – 7.3x) = 16.4x + 5.8b) 6.6(1.2 – 7.3x) = 16.4x + 5.8

Example 2Example 2: : Multiply the equation Multiply the equation by a power of 10 to write an by a power of 10 to write an equivalent equation with integer equivalent equation with integer coefficients. Solve the equation and coefficients. Solve the equation and round to the nearest hundredth.round to the nearest hundredth.

a) 3.11x – 17.64 = 2.02x -5.89a) 3.11x – 17.64 = 2.02x -5.89

b) 5.8 + 3.2x = 3.4x – 16.7b) 5.8 + 3.2x = 3.4x – 16.7

3.7 Formulas and Functions3.7 Formulas and Functions

Objective: To solve a formula for one Objective: To solve a formula for one of its variables and rewrite an of its variables and rewrite an equation in function formequation in function form

FormulaFormula: : an algebraic expression an algebraic expression that relates two or more real-life that relates two or more real-life quantities.quantities.

Example 1Example 1: : Use the formula for Use the formula for area of a rectangle – area of a rectangle – A = lwA = lw

a) Find a formula for a) Find a formula for ll in terms of in terms of A A and and ww

b) Use the new formula to find the b) Use the new formula to find the length of a rectangle that has an length of a rectangle that has an area of 35 sq. ft. and a width of 7 area of 35 sq. ft. and a width of 7 feet.feet.

Example 2Example 2: : Solve the temperature Solve the temperature formula formula C C = 5/9(F – 32)= 5/9(F – 32) for for FF..

Example 3Example 3: : a) Solve the simple a) Solve the simple interest formula for interest formula for rr..

b) Find the interest rate for an b) Find the interest rate for an investment of $1500 that earned $54 investment of $1500 that earned $54 in simple interest in one yearin simple interest in one year

Example 3Example 3: : Rewrite the equation Rewrite the equation

3x + y = 4 so that y is a function of x.3x + y = 4 so that y is a function of x.

Example 4Example 4: : a) Rewrite the equationa) Rewrite the equation

3x + y = 4 so that x is a function of y.3x + y = 4 so that x is a function of y.

b) Use the result to find x when y = -b) Use the result to find x when y = -2, -1, 0 and 12, -1, 0 and 1