Section 6.2 Solving Linear Equations Math in Our World

Learning Objectives  Decide if a number is a solution of an equation.  Identify linear equations.  Solve general linear equations.  Solve linear equations containing fractions.  Solve formulas for one specific variable.  Determine if an equation is an identity or a contradiction.

Equations An equation is a statement that two algebraic expressions are equal. A solution of an equation is a value of the variable that makes the equation a true statement when substituted into the equation. Solving an equation means finding every solution of the equation. We call the set of all solutions the solution set, or simply the solution of an equation. For example, x = 2 is one solution of the equation x 2 – 4 = 0, because (2) 2 – 4 = 0 is a true statement. But x = 2 is not the solution, because x = – 2 is a solution as well. The solution set is actually {– 2, 2}.

Expressions vs. Equations Note the difference between the two; equations contain an equal sign and expressions do not.

EXAMPLE 1 Identifying Solutions of an Equation Determine if the given value is a solution of the equation. (a) 4(x – 1) = 8; x = 2 (b) x + 7 = 2x – 1; x = 8 (c) 2y 2 = 200; y = – 10

EXAMPLE 1 Identifying Solutions of an Equation (a)Substituting 2 in for x we get: This is not true, so x = 2 is not a solution of the equation. (b) Substituting 8 in for x we get: This is true, so x = 8 is a solution of the equation. SOLUTION

EXAMPLE 1 Identifying Solutions of an Equation (c) Substituting in – 10 for y, we get: This is true, so y = – 10 is a solution of the equation. SOLUTION

Linear Equations A linear equation in one variable is an equation that can be written in the form Ax + B = 0, where A and B are real numbers, and A is not zero. For example, 2x + 3 = 7 is a linear equation.

EXAMPLE 2 Identifying Linear Equations Determine which of the equations below are linear equations. (a) This is a linear equation: the variable appears only to the first power. (b) This is not a linear equation: the variable has exponent 2. (c) This is a linear equation. Don’t let the fraction, or the exponent on 3 fool you! The variable has exponent 1. (d) This is not a linear equation. The left side can be simplified to x 2 + 3x, so we see that x has exponent two. SOLUTION

Solving Linear Equations Addition and Subtraction Properties of Equality You can add or subtract the same real number or algebraic expression to both sides of an equation without changing the solution set. In symbols, if a = b, then a + c = b + c and a – c = b – c. Multiplication and Division Properties of Equality You can multiply or divide both sides of an equation by the same nonzero real number without changing the solution set. In symbols, if a = b, then ac = bc and ac bc as long as c ≠ 0.

EXAMPLE 3 Solving Linear Equations Solve each equation using the Addition and Subtraction Property, and check your answer. (a) x – 5 = 9 (b) y + 30 = 110

EXAMPLE 3 Solving Linear Equations (a) The goal is to transform the equation into one whose solution is obvious. We can do this by isolating the variable, x, on the left side. To do so, we will add 5 to both sides. The solution set is now obvious: it’s {14}. One of the great things about solving equations is that you can easily check your answer by substituting back into the original equation: SOLUTION

EXAMPLE 3 Solving Linear Equations (b) This time, to isolate the variable, we subtract 30 from both sides. The solution set is {80}. SOLUTION

EXAMPLE 4 Solving Linear Equations Solve each equation using the Multiplication and Division Property, and check your answer.

(a) This time, to isolate the variable, we need to multiply both sides of the equation by 6. The solution set is {18}. EXAMPLE 4 Solving Linear Equations SOLUTION

(b) The variable is multiplied by 5, so we can isolate x by dividing both sides by 5. EXAMPLE 4 Solving Linear Equations SOLUTION

Solving Linear Equations Procedures for Solving Linear Equations Step 1 Simplify the expressions on both sides of the equation by distributing and combining like terms. Step 2 Use the addition and/or subtraction property of equality to move all the variable terms to one side of the equation and all the constant terms to the opposite side of the equation. Step 3 Combine like terms. Step 4 Use the multiplication or division property of equality to eliminate the numerical coefficient and solve for the variable.

EXAMPLE 5 Solving a Linear Equation Solve the equation 5x + 9 = 29. SOLUTION

EXAMPLE 6 Solving a Linear Equation SOLUTION Solve the equation 6x – 10 = 4x + 8.

EXAMPLE 7 Solving a Linear Equation Solve the equation 3(2x + 5) – 10 = 3x – 10.

EXAMPLE 7 Solving a Linear Equation SOLUTION

Solving Equations Containing Fractions There’s a simple procedure that will turn any equation with fractions into one with no fractions at all. You just need to find the Least Common Denominator of all fractions that appear in the equation, and multiply every single term on each side of the equation by the LCD. If there are any fractions left after doing so, you made a mistake!

EXAMPLE 8 Solving Linear Equations Containing Fractions Solve the equation:

The LCD of fractions with denominators 3 and 5 is 15. So the first step is to multiply every term in the equation by 15. EXAMPLE 8 Solving Linear Equations Containing Fractions SOLUTION

EXAMPLE 10 Solving a Formula in Electronics for One Variable Solve the formula SOLUTION

EXAMPLE 11 Finding a Formula for Temperature in Celsius The formula F = 95 C + 32 gives the Fahrenheit equivalent for a temperature in Celsius. Transform this into a formula for calculating the Celsius temperature C.

EXAMPLE 11 Finding a Formula for Temperature in Celsius SOLUTION

Contradictions and Identities A contradiction is an equation with no solution. An identity is an equation that is true for any value of the variable for which both sides are defined. When you solve an equation that is an identity, the final equation will be a statement that is always true. In a contradiction the final equation will be a statement that is false.

EXAMPLE 12 Recognizing Identities and Contradictions Indicate whether the equation is an identity or a contradiction, and give the solution set. (a) 3(x – 6) + 2x = 5x – 18 (b) 6x – 4 + 2x = 8x – 10

EXAMPLE 12 Recognizing Identities and Contradictions SOLUTION Since the resulting equation is true, it is an identity and the solution set is {x | x is a real number}.

EXAMPLE 12 Recognizing Identities and Contradictions SOLUTION Since the resulting equation is false, it’s a contradiction, and the solution set is .

EXAMPLES Sample Exercises 13. 7(x – 3) = 2x is Linear 19. 3(x – 2) 2 = 12; x = 0 3(-2) 2 = 12 3(4) = = 12 YES this is a solution x – 7y = 18 5x – 7y = 16 5x = 7y + 16 x = 7y