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: 2.8.11 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 – 3Addition Property of EqualityDivision Property of Equality

2). x = x + 22 2

Multiplication Property of EqualitySubtraction 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 + 32. Find the Celsius temperature that is equivalent to 86 F. 86 = 9/5C + 32

86 – 32 = 9/5C54 = 9/5C

30 = C

II. Substitution PropertyUsing the equation given in Example 1, find the Celsius temperature that is equivalent to 122 F.

122 = 9/5C + 32122 – 32 = 9/5C

90 = 9/5CC = 50

Solve 3x – 8 = 5x – 20. Check your solution by using substitution.

3x – 8 = 5x - 20-2x – 8 = -20

-2x = -12X = 6

Check the solution by substitution:3(6) – 8 = 5 (6) – 20 18 – 8 = 30 – 20 10 = 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 + 3.65 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 -8.76X = 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 EquationsEx 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 = 2Ab = 2A/h

Ex 2. P = 2l + 2w for w P – 2l = 2w(P-2l)/2 = w

V. Literal Equations

Ex 3. A = ½ h(b1 + b2) for b22A = h (b1 + b2)

(2A)/h = (b1 + b2)b2 = (2A)/h – b1

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