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 – 3

Addition Property of Equality

Division Property of Equality

2). x = x + 22 2

Multiplication Property of Equality

Subtraction Property of Equality

II. Substitution Property If a = b, you may replace a with b.

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.

II. Substitution Property

Using the equation given in Example 1, find

the Celsius temperature that is equivalent to

122 F.

Solve 3x – 8 = 5x – 20.

Check your solution by using substitution.

Check the solution by substitution:

Solve 7 – 6x = 2x –9. Check your solution by using substitution.

III. An equation may also be solved by

graphing!! Type it in y =. Use trace to find the point.

Ex 1. Solve 3.24x – 4.09 = -0.72x + 3.65 by graphing.

Type into your graphing calculator:

Left side of equation: Right side of equation:y1 = 3.24x – 4.09 y2 = -0.72x + 3.65

III. An equation may also be solved by graphing!!

Type it in y =. Use trace to find the point.

Ex 2. Solve 2.24x – 6.24 = 4.26x – 8.76 by graphing.

Left side of equation: Right side of equation:

y1 = 2.24x – 6.24 y2 = 4.26x -8.76

x = 1.25

IV. Solve Multi-Step Equations

Simplify each side of the equation Distribute Combine Like Terms

Add/subtract the smallest variable term (if there are variables on both sides)

Solve the resulting one or two step

equation

IV. Solve Multi-Step EquationsEx 1. –2x –7 = 9

Ex 2. 4x + 80 = -6x

Ex 3. 3x – 8 = 2x + 2

V. Literal Equations

An equation that contains two or more variables.

Formulas are examples of literal equations.

Ex 1. ½ bh = A for b

Ex 2. P = 2l + 2w for w

V. Literal Equations

Ex 3. A = ½ h(b1 + b2) for b2

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

Integrated Algebra II – Section 1.6 Level A

Academic Algebra II – Section 1.6 Level B