1 MATH 131 Unit 1 Prof. Tereba Lesson 1 LF 01 Linear Equations In this lesson , you will: • Solve equations in one variable algebraically . Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. Solving an equation means “undo” all the operations in the equation, leaving the variable y by itself on one side. This is known as isolating the variable. 1. General Equations Remember that you can think of an equation as a balance scale, aiming to rewrite it so it is easier to solve while remaining balanced. General equations In general, there is a 5 -step process to solving any linear equation. While all five steps aren’t always needed, this can serve as a guide for solving equations. Step 1. Apply the distributive property to rewrite the equation without parentheses . Step 2. Combine like terms on each side of the equation. Step 3. Isolate the variable term by applying the addition property. Step 4 . Solve the equation by applying the multiplication property. Step 5 . ✓Verify the solution. The Distributive Property of Multiplication For all real numbers a, b, and c, a (b + c) = a ∙ b + a ∙ c What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. LINEAR EQUATION IN ONE VARIABLE . A linear equation in one variable can be written in the form : ax + b = 0 where a and b are real numbers, a≠0 . A linear equation of one variable is an equation with one variable with exponent one. 2 Example 1: Solve the equation by using the general strategy. a) 6(3y + 2) − 8 = − 2 b) 12−3(x−1)=−5(x+2)+7x A 2. Solving Equations with Fractions Clearing Denominators We can easily clear denominators in an equation by multiplying each term by the LCD. After completing this step, the fractions are cleared, and we can work with a more familiar type of equation. Example 2: Solve the equation for the given variable. a) x+8 4= −5 b) 5 3x+2 = −5+1 6𝑥 3 c) 3 5n−7 10=−4+7 15n 3. Solutions to Linear Equations The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for x will make the equation true. A conditional equation is true for only some values of the variable. An identity equation is true for all values of the variable. An inconsistent equation results in a false statement. Solutions to linear equations can fall into three categories: • One solution • No solution, DNE (does not exist) • Many solutions, also called infinitely many solutions or All Real Numbers Example 3: Solve the equation for the given variable. a) 3 (2x – 5) = 6x − 15 b) 5x + 3 – 4x = 2 + x