1. 1.2 Linear Equations and Rational Equations

2. Terms Involving Equations 3x - 1 = 2 An equation consists of two algebraic expressions joined by an equal sign. 3x – 1 = 2 3x = 3 x = 1 1 is a solution or root of the equation. If you have a SOLUTION to an equation, that is the value for the variable(s) that make the equation “true”. Left Side Right Side

3. Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the form ax + b = 0 where a and b are real numbers and a ≠ 0.

4. An equation can be transformed into an equivalent equation by one or more of the following operations. 1. S IMPLIFY each side by removing grouping symbols and combining like terms. 3(x - 6) = 6x - x 3x - 18 = 5x -18 = 2x -9 = x Divide both sides of the equation by 2. 3. D IVIDE (or multiply)on both sides of the equation by the same nonzero quantity. Subtract 3x from both sides of the equation. 3x - 18 = 5x 3x - 18 - 3x = 5x - 3x -18 = 2x 2. A DD (or subtract) the same real number or variable expression on both sides of the equation to get the variable on one side, and constants on the other. -9 = x x = -9 4. Write your final solution (for a linear equation with one variable) with the variable on the left. Generating Equivalent Equations

5. Solving a Linear Equation OPTIONAL: multiply through by LCD or multiple of 10 to clear fractions or decimals SIMPLIFY the algebraic expression on each side. ADD to collect all the variable terms on one side and all the constant terms on the other side. DIVIDE by the coefficient of the variable to isolate the variable. Check the proposed solution in the original equation.

6. Solve the equation: 2(x - 3) - 17 = 13 - 3(x + 2). Solution Step 1 SIMPLIFY the algebraic expression on each side. 2(x - 3) – 17 = 13 – 3(x + 2) This is the given equation. Ex. Step 2ADD to collect variable terms on one side and constant terms on the other side. Step 3DIVIDE by the coefficient of the variable to isolate the variable and solve.

7. The solution set is {____}. Step 4 Check the proposed solution in the original equation. Substitute 6 for x in the original equation. 2(x - 3) - 17 = 13 - 3(x + 2) This is the original equation. Substitute 6 for x. Con’t. ?

8. Types of Equations Identity:An equation that is true for all real numbers. (0 = 0, all real numbers) Conditional:An equation that is true for at least one real number. (x = 0, or any constant) Inconsistent:An equation that is not true for any real number. (0 = 5, NO SOLUTION)

9. Determine whether the equation 3(x - 1) = 3x + 5 is an identity, a conditional equation, or an inconsistent equation. Solution To find out, solve the equation. 3(x – 1) = 3x + 5 This equation is _________________. Example Do p 104 # 27 & 49 in class.