Download 1.6 Introduction to Solving Equations

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 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 = 9/5C
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
122 – 32 = 9/5C
90 = 9/5C
C = 50
Solve 3x – 8 = 5x – 20.
Check your solution by using substitution.
3x – 8 = 5x - 20
-2x – 8 = -20
-2x = -12
X=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 -8.76
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(b1 + b2) for b2
2A = 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.