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1.2 Linear Equations and Rational Equations Terms Involving Equations 3x - 1 = 2 Left Side Right Side 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”. 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. Generating Equivalent Equations An equation can be transformed into an equivalent equation by one or more of the following operations. 1. SIMPLIFY each side by removing grouping symbols and combining like terms. 3(x - 6) = 6x - x 3x - 18 = 5x 2. ADD (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. 3x - 18 = 5x 3x - 18 - 3x = 5x - 3x -18 = 2x 3. DIVIDE (or multiply)on both sides of the equation by the same nonzero quantity. -18 = 2x -9 = x 4. Write your final solution (for a linear equation with one variable) with the variable on the left. -9 = x x = -9 Subtract 3x from both sides of the equation. Divide both sides of the equation by 2. 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. Ex. Solve the equation: 2(x - 3) - 17 = 13 - 3(x + 2). Solution 2(x - 3) – 17 = 13 – 3(x + 2) This is the given equation. Step 1 SIMPLIFY the algebraic expression on each side. Step 2 ADD to collect variable terms on one side and constant terms on the other side. Step 3 DIVIDE by the coefficient of the variable to isolate the variable and solve. Con’t. 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. The solution set is {____}. 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) Example 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 _________________. Do p 104 # 27 & 49 in class.