two equations are equivalent if they have the same solutions. solving a linear equation an equation...
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Two equations are equivalent if they have the same solutions.
Solving a Linear Equation
An equation is a statement in which two expressions are equal.
A linear equation in one variable is an equation that can be written in the form ax = b where a and b are constants and a 0.
A number is a solution of an equation if the statement is true when the number is substituted for the variable.
For instance, the equations x – 4 = 1 and x = 5 are equivalent because both have the number 5 as their only solution.
The transformations, or changes, on the following slide produce equivalent equations and can be used to solve an equation.
Solving a Linear Equation
TRANSFORMATIONS THAT PRODUCE EQUIVALENT EQUATIONS
ADDITION PROPERTY
OF EQUALITY
SUBTRACTION PROPERTY
OF EQUALITY
MULTIPLICATION PROPERTY
OF EQUALITY
DIVISION PROPERTY
OF EQUALITY
Add the same number to both sides:
If a = b, then a + c = b + c.
Subtract the same number from both sides:
If a = b, then a – c = b – c.
Multiply both sides by the same nonzero
number: If a = b and c 0, then ac = bc.
Divide both sides by the same nonzero
number: If a = b and c 0, then a c = b
c.
Your goal is to isolate the variable on one side of the equation.
Write original equation.
Subtract 9 from each side.
Simplify.
SOLUTION
Solving an Equation with a Variable on One Side
Solve + 9 = 15.37
x
+ 9 = 1537
x
= 6 37
x
Multiply each side by , the reciprocal of . 73
37 = 6 x
73
(6)
= 6 x 14
The solution is 14. Substitute 14 for x.
Solution checks.
+ 9 = 1537
( x )
CHECK
?
15 = 15
14
Example
• Fractions (Multiply by LCD)
3
4
4
52
11
4x F
HGIKJ
You do: (multiply by LCD)
3 2 5
4 3 6x
Solve 5n + 11 = 7n – 9.
5n + 11 = 7n – 9
5n + 11 = 2n – 9
5n + 20 = 2n
10 = n Divide each side by 2.
Add 9 to each side.
Subtract 5n from each side.
Write original equation.
SOLUTION
Solving an Equation with a Variable on Both Sides
The solution is 10. Check this in the original equation.
You do:
6x – 4 = 2x + 10
Solve 4(3x – 5) = –2(–x + 8) – 6x.
4(3x – 5) = –2(–x + 8) – 6x
12x – 20 = 2x –16 – 6x
12x – 20 = – 4x – 16
Divide each side by 16.
Combine like terms.
Distributive property
Write original equation.
SOLUTION
Using the Distributive Property
16x – 20 = –16 Add 4x to each side.
16x = 4 Add 20 to each side.
x =14
The solution is . Check this in the original equation.14
You do:
2(x + 2) = 3(x – 8)
Example• Interesting:
3 8 3 24
5 12 6 12
( )x x
x x x
REAL ESTATE A real estate broker’s base salary is $18,000. She earns a 4%
commission on total sales. How much must she sell to earn $55,000 total?
SOLUTION
Verbal Model
Labels
Total income
Commissionrate
Total income = 55,000
Base salary = 18,000
Commission rate = 0.04
(dollars)
(dollars)
(percent in decimal form)
Basesalary
+
Using Linear Equations in Real Life
= •Totalsales
Total sales = x (dollars)
AlgebraicModel 18,000 + 0.04x55,000 =
REAL ESTATE A real estate broker’s base salary is $18,000. She earns a 4%
commission on total sales. How much must she sell to earn $55,000 total?
SOLUTION
18,000 + 0.04x55,000 =
37,000 = 0.04x
925,000 = x
Write linear equation.
Subtract 18,000 from each
side.
Divide each side by 0.04.
The broker must sell real estate worth a total of $925,000 to earn $55,000.
Writing and Using a Geometric Formula
You do:
A car salesperson’s base salary is $21,000. She earns a 5 % commission on sales. How much must she sell to earn $65,000 total?
Example• The state sales tax in Pennsylvania is 0.06
(or 6%). If your total bill at the music store included $1.32 in tax, how much did the merchandise cost?
You have a 3 inch by 5 inch photo that you want to enlarge, mat, and frame.
You want the width of the mat to be 2 inches on all sides. You want the
perimeter of the framed photo to be 44 inches. By what percent should you
enlarge the photo?
SOLUTION
Verbal Model
Labels
Perimeter +
Perimeter = 44 (inches)
(inches)
(inches)
2 • Width= 2 • Length
Width = 4 + 3x
Length = 4 + 5x
44 = 2(4 + 3x) + 2(4 + 5x)Algebraic
Model
Writing and Using a Geometric Formula
44 = 2(4 + 3x) + 2(4 + 5x) Write linear equation.
Distribute and combine like terms.44 = 16 + 16x
Subtract 16 from each side.28 = 16x
Divide each side by
16.
1.75 = x
You have a 3 inch by 5 inch photo that you want to enlarge, mat, and frame.
You want the width of the mat to be 2 inches on all sides. You want the
perimeter of the framed photo to be 44 inches. By what percent should you
enlarge the photo?
SOLUTION
You should enlarge the photo to 175% of its original size.
Writing and Using a Geometric Formula