College Algebra Chapter 1 Equations and Inequalities Section 1.1 Linear Equations and Rational Equations

1. Solve Linear Equations in One Variable 2. Identify Conditional Equations, Identities, and Contradictions 3. Solve Rational Equations 4. Solve Literal Equations for a Specified Variable

Solve Linear Equations in One Variable Solving a Linear Equation in One Variable Step 1 Simplify both sides of the equation.   Use the distributive property to clear parentheses. Combine like terms. Consider clearing fractions or decimals by multiplying both sides of the equation by the least common denominator (LCD) of all terms.

Solve Linear Equations in One Variable Solving a Linear Equation in One Variable Step 2 Use the addition property of equality to collect the variable terms on one side of the equation and the constant terms on the other side. Step 3 Use the multiplication property of equality to make the coefficient on the variable term equal to 1. Step 4 Check the potential solution in the original equation. Step 5 Write the solution set.

Example 1: Solve:

Example 2: Solve:

Example 3: Solve:

Example 4: Solve:

1. Solve Linear Equations in One Variable 2. Identify Conditional Equations, Identities, and Contradictions 3. Solve Rational Equations 4. Solve Literal Equations for a Specified Variable

Identify Conditional Equations, Identities, and Contradictions A conditional equation is true for some values of the variable and false for other values.   An identity is an equation that is true for all values of the variable for which the expressions in an equation are defined. A contradiction is an equation that is false for all values of the variable.

Example 5: Solve:

Example 6: Solve:

Example 7: Solve:

1. Solve Linear Equations in One Variable 2. Identify Conditional Equations, Identities, and Contradictions 3. Solve Rational Equations 4. Solve Literal Equations for a Specified Variable

Solve Rational Equations A rational equation is an equation in which each term contains a rational expression. All linear equations are rational equations, but not all rational equations are linear. When a variable appears in the denominator of a fraction, we must restrict the values of the variable to avoid division by zero.

Example 8: restricted value for x : ________ Solve:

Example 9: restricted value for x : ________ Solve:

Example 10: restricted values for y: ________ Solve:

1. Solve Linear Equations in One Variable 2. Identify Conditional Equations, Identities, and Contradictions 3. Solve Rational Equations 4. Solve Literal Equations for a Specified Variable

Example 11: Solve for x.

Example 12: Solve for y.

Example 13: Solve for z.

Example 14: Solve for x.

Example 15: Adele's 2012 Mini-Cooper gets 37 miles per gallon on the highway and 29 miles per gallon in the city. The amount of gas she uses A (in gallons) is given by where c is the number of city miles driven and h is the number of highway miles driven. If Adele drove 58 miles in the city and used 5 gallons of gas, how many highway miles did she drive?

Example 15 continued: If Adele drove 58 miles in the city and used 5 gallons of gas, how many highway miles did she drive?