Section 6.2 Linear Equations in One Variable

What You Will Learn Linear Equations Solving Linear Equations

Definitions Terms are parts that are added or subtracted in an algebraic expression. 4x – 3y – 5 has three terms. Numerical Coefficient or Coefficient is the numerical part of a term. The coefficient of 4x is 4, –3y is –3. –5 is considered a constant.

Definitions Like terms are terms that have the same variables with the same exponents on the variables. Unlike terms have different variables or different exponents on the variables.

Simplify To simplify an expression means to combine like terms by using the commutative, associative, and distributive properties.

Properties of the Real Numbers Associative property of multiplication (ab)c = a(bc) Associative property of addition (a + b) + c = a + (b + c) Commutative property of multiplication ab = ba Commutative property of addition a + b = b + a Distributive property a(b + c) = ab + ac

Example 1: Combining Like Terms Combine like terms in each expression.

Definitions Linear (or first-degree) equation in one variable is one in which the exponent on the variable is 1. Equivalent equations are equations that have the same solution.

Solving Equations To solve any equation, isolate the variable. That means getting the variable by itself on one side of the equal sign. The four properties of equality that we are about to discuss are used to isolate the variable. The first is the addition property.

Addition Property of Equality If a = b, then a + c = b + c for all real numbers a, b, and c. The addition property of equality indicates that the same number can be added to both sides of an equation without changing the solution.

Example 2: Using the Addition Property of Equality Determine the solution to the equation x – 9 = 15. Solution

Subtraction Property of Equality If a = b, then a – c = b – c for all real numbers a, b, and c. The subtraction property of equality indicates that the same number can be subtracted from both sides of an equation without changing the solution.

Example 3: Using the Subtraction Property of Equality Determine the solution to the equation x + 11 = 19. Solution

Multiplication Property of Equality If a = b, then a • c = b • c for all real numbers a, b, and c, where c ≠ 0. The multiplication property of equality indicates that both sides of the equation can be multiplied by the same nonzero number without changing the solution.

Example 4: Using the Multiplication Property of Equality Determine the solution to Solution

Division Property of Equality If a = b, then for all real numbers a, b, and c, where c ≠ 0. The division property of equality indicates that both sides of an equation can be divided by the same nonzero number without changing the solution.

Example 5: Using the Division Property of Equality Determine the solution to Solution

General Procedure for Solving Linear Equations 1. If the equation contains fractions, multiply both sides of the equation by the lowest common denominator (or least common multiple). This step will eliminate all fractions from the equation.

General Procedure for Solving Linear Equations 2. Use the distributive property to remove parentheses when necessary. 3. Combine like terms on the same side of the equal sign when possible.

General Procedure for Solving Linear Equations 4. Use the addition or subtraction property to collect all terms with a variable on one side of the equal sign and all constants on the other side of the equal sign. It may be necessary to use the addition or subtraction property more than once. This process will eventually result in an equation of the form ax = b, where a and b are real numbers.

General Procedure for Solving Linear Equations 5. Solve for the variable using the division or multiplication property. The result will be an answer in the form x = c, where c is a real number.

Example 8: Solving an Equation Containing Fractions Solve the equation Solution

Example 8: Solving an Equation Containing Fractions Solution

Example 9: Variables on Both Sides of the Equation Solve the equation Solution

Example 12: An Equation with No Solutions Solve Solution False

Example 12: An Equation with No Solutions During the process of solving an equation, if you obtain a false statement like –3 = –8, or 5 = 0, the equation has no solution. An equation that has no solution is called a contradiction. The equation 2(x – 4) + x + 5 = 5x – 2(x + 4) is a contradiction and thus has no solution.

Example 13: An Equation with Infinitely Many Solutions Solve Solution At this point both sides of the equation are the same. Every real number will satisfy this equation. This equation has an infinite number of solutions.

Example 13: An Equation with Infinitely Many Solutions An equation of this type is called an identity. The solution to any linear equation with one variable that is an identity is all real numbers. If you continue to solve an equation that is an identity, you will end up with 0 = 0, as follows.

Example 13: An Equation with Infinitely Many Solutions

Ratios A ratio is a quotient of two quantities. For example, 2 to 5, which can be written 2:5 or 2/5 or

Proportions A proportion is a statement of equality between two ratios.

Cross Multiplication Proportions can be solved using cross multiplication. If then ad = bc, b ≠ 0, d ≠ 0.

To Solve Application Problems Using Proportions 1. Represent the unknown quantity by a variable. 2. Set up the proportion by listing the given ratio on the left-hand side of the equal sign and the unknown and other given quantity on the right-hand side of the equal sign.

To Solve Application Problems Using Proportions When setting up the right-hand side of the proportion, the same respective units should occupy the same respective positions on the left and right. For example, an acceptable proportion might be

To Solve Application Problems Using Proportions 3. Once the proportion is properly written, drop the units and use cross multiplication to solve the equation. 4. Answer the question or questions asked using the appropriate units.

Example 14: Water Usage The cost for water in the city of Tucson, Arizona is $1.75 per 750 gallons (gal) of water used. What is the water bill if 30,000 gallons are used? Solution

Example 14: Water Usage Solution 30,000 gallons of water costs $750.