1.6 Introduction to Solving Equations Objectives: Write and solve a linear equation in one variable. Solve a literal equation for a specified variable. Standard: D Formulate equations to model routine and non-routine problem.

An equation is a statement that two expressions are equal. A variable is a symbol that represents many different numbers in a set of numbers. Any value of a variable that makes an equation true is a solution of the equation.

I. Properties of Equality For real numbers a, b, c: Reflexive Property a = a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Addition Property If a = b, then a + c = b + c. Subtraction Property If a = b, then a – c = b – c. Multiplication Property If a = b, then ac = bc. Division Property If a = b, then a  c = b  c, c  0.

I. Properties of Equality Tell which Properties of Equality you would use to solve each equation. 1). 52 = -2.7x – 3 Addition Property of Equality Division Property of Equality 2). x = x Multiplication Property of Equality Subtraction Property of Equality

II. Substitution Property If a = b, you may replace a with b in any true statement containing a and the resulting statement will still be true. Ex 1. The relationship between the Celsius temperature, C, and the Fahrenheit temperature, F, is given by F = 9/5 C Find the Celsius temperature that is equivalent to 86  F. 86 = 9/5C – 32 = 9/5C 54 = 9/5C 30 = C

II. Substitution Property Using the equation given in Example 1, find the Celsius temperature that is equivalent to 122  F. 122 = 9/5C – 32 = 9/5C 90 = 9/5C C = 50

Solve 3x – 8 = 5x – 20. Check your solution by using substitution. 3x – 8 = 5x x – 8 = x = -12 X = 6 Check the solution by substitution: 3(6) – 8 = 5 (6) – – 8 = 30 – = 10

Solve 7 – 6x = 2x –9. Check your solution by using substitution. 7 – 6x = 2x – 9 -8x = -16 X = 2 Check the solution by substitution: 7 – 6(2) = 2(2) – 9 7 – 12 = 4 – 9 -5 = -5

III. An equation may also be solved by graphing!! Type it in y =. Trace to find the point. Ex 1. Solve 3.24x – 4.09 = -0.72x by graphing.

III. An equation may also be solved by graphing!! Type it in y =. Trace to find the point. Ex 2. Solve 2.24x – 6.24 = 4.26x – 8.76 by graphing. Y = 2.24x – 6.24 and y = 4.26x X = 1.25

IV. Solve Multi-Step Equations Distribute Combine Like Terms Bring Letters to the Left Bring Numbers to the Right Solve for the variable

IV. Solve Multi-Step Equations Ex 1. –2x –7 = 9 -2x = 16 x = -8 Ex 2. 4x + 80 = -6x 10x = -80 x = -8 Ex 3. 3x – 8 = 2x + 2 x – 8 = 2 x = 10

V. Literal Equations An equation that contains two or more variables. Formulas are examples of literal equations. Ex 1. ½ bh = A for b bh = 2A b = 2A/h Ex 2. P = 2l + 2w for w P – 2l = 2w (P-2l)/2 = w

V. Literal Equations Ex 3. A = ½ h(b 1 + b 2 ) for b 2 2A = h (b 1 + b 2 ) (2A)/h = (b 1 + b 2 ) b 2 = (2A)/h – b 1

Writing Activities: Solving Equations 9). Solve 5x – 1 = 3x – 15. Explain each step, and include the Properties of Equality that you used. 10). Explain how you can verify that 3(2x + 5) = 9 + 3x and x = -2 are equivalent equations.

Homework Pg. 49 #12 – 60 even