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6.2 Linear Equations in One Variable and Proportions

Ch. 6Equations and Inequalities ObjectivesSolve linear equations.Solve linear equations containing fractions.Solve proportions.Solve problems using proportions.Identify equations with no solution or infinitely many solutions.

Linear Equationslinear equation in one variable x is an equation that can be written in the formax + b = 0, where a and b are real numbers, and a 0.

Solving an equation in x determining all values of x that result in a true statement when substituted into the equation.

Equivalent equations have the same solution set. 4x + 12 = 0 and x = 3 are equivalent equations.

3Addition and Multiplying PropertiesThe Addition Property of EqualityIf a = b, then a + c = b + cThe Multiplication Property of EqualityIf a = b, then a c = b c

Using Properties of Equality to Solve Equations

Solving Linear EquationSolve and check: 2(x 4) 5x = 5.Step 1. Simplify the algebraic expression on each side:2(x 4) 5x = 52x 8 5x = 5. 3x 8 = 5Step 2. Collect variable terms on one side and constants on the other side.3x 8 + 8 = 5 + 83x = 3

Solving Linear EquationStep 3. Isolate the variable and solve. 3x = 3 3 3 x = 1Step 4. Check the proposed solution in the original equation by substituting 1 for x. 2(x 4) 5x = 52(1 4) 5(1) = 510 (5) = 55 = 5

Because the check results in a true statement, we conclude that the solution set of the given equation is {1}.

Application

These graphs indicate that persons with a low sense of humor have higher levels of depression. These graphs can be modeled by the following formulas:

We are interested in the intensity of a negative life event with an average level of depression of 7/2 for the high humor group.

Linear Equation with No SolutionSolve: 2x + 6 = 2(x + 4)Solution:2x + 6 = 2(x + 4)2x + 6 = 2x + 8 2x + 6 2x = 2x + 8 2x 6 = 8The original equation 2x + 6 = 2(x + 4) is equivalent to 6 = 8, which is false for every value of x. The equation has no solution. The solution set is .

Linear Equation with Infinitely Many solutionsSolve: 4x + 6 = 6(x + 1) 2x

Solution:4x + 6 = 6(x + 1) 2x4x + 6 = 6x + 6 2x4x + 6 = 4x + 6The original statement is equivalent to the statement 6 = 6, which is true for every value of x. The solution set is the set of all real numbers, expressed as {x|x is a real number}.

ProportionsThe property tax on a house with an assessed value of $480,000 is $5760. Determine the property tax on a house with an assessed value of $600,000, assuming the same tax rate.

5760 x ------------ = ------------ 480,000 600,000

5760 x ------------ = ------------ 480,000 600,000 5760 x ----------- = -------- 48 60 5760 60 ------------- = x 48 5 1440 155760 60------------ = x 48 12 4 17200 = x$7200 = xExample: Changing Recipe SizeA chocolate-chip recipe for five dozen cookies requires cup of sugar. If you want to make eight dozen cookies, how much sugar is needed?Solution: 3 / 4 (cup) x (cup)----------------- = -------------- 5 (dozen) 8 (dozen)

8 (3/4) 8 3 6---------- = x --- --- = --- = x 5 5 4 5x60240960Find the height x of the tree, when a 60-man casts a shadow 240 long.Proportionx60240960 60 x ----- = ------- 240 960

60 960 ----------- = x x = 240 240