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Study Guides index of guidesSolving Linear Equations

 

The order and connection of ideas
is the same as the order
and connection of things.
Baruch (Benedict) Spinoza
1623-1677, Dutch

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Definitions:
Linear
Equation:		  a mathematical expression that has an equal sign and linear expressions
Variable: a number that you don't know,
often represented by "x" or "y" but any letter will do!
Linear
Expression:

a mathematical statement that performs functions of
addition, subtraction, multiplication, and division
Note:  Variable(s) in linear expressions

  • Cannot have exponents (or powers)
    For example, x squared or x 2

  • Cannot multiply or divide each other
    For example:  "x" times "y" or xy; "x" divided by "y" or x/y

  • Cannot be found under a root sign or square root sign (sqrt)
    For example:  Ö x or the "square root of x"; sqrt (x)
These are examples of linear expressions:
x + 4
2x + 4
2x + 4y
These are not linear expressions:
x2 (no exponents on variables)
2xy + 4 (can't multiply two variables)
2x / 4y (can't divide two variables)
Ö x (no square root sign on variables)

Solve these linear equations by clicking and dragging
a number to the "other" side of the equal sign. 
Remember that you are "isolating" the unknown "X" to solve the problem. 
(Examples are provided below.)

Solving example #1:  find x if  2x + 4 =  10

  linear equation #1:
steps to solve

2x + 4 =  10
math

1.

 Isolate "x" to one side of the equation
by subtracting 4 from both sides:		 2x + 4 - 4 = 10 - 4

2x = 6
2. Divide both sides by 2: 2x  / 2 = 6 / 2

x = 3
3. Check your work with the original equation:  2x + 4 = 10
(2 * 3) + 4 = 10
6 + 4 = 10

Solving example #2:  find x if  3x - 4 = -10 
(using negatives)

  linear equation #2:
steps to solve

3x - 4 = -10
math

1.

 Isolate "x" to one side of the equation
by adding 4 to both sides:		 3x - 4 + 4 = -10 + 4

3x = -6
2. Divide both sides by 3: 3x / 3 = -6 / 3
x = -2
3. Check your work with the original equation: (3 * -2) - 4 = -10
-6 - 4 = -10

Solving example #3: find x if: 4x - 4y = 8 
(using more than one variable)

  linear equation #3:
steps to solve		 4x - 4y = 8
math

1.

 First step is to isolate "x" to one side of the equation
by adding 4y to both sides:		 4x - 4y + 4y = 8 + 4y
4x = 8 + 4y
2. Second step is to divide both sides by 4: 4x / 4 = (8 + 4y) / 4
x = 2 + y
3. Check your work with the original equation: 4 * (2 + y) - 4y = 8
8 + 4y - 4y = 8
8 = 8

Solving example #4:  find x if:  x + 32 = 12

  linear equation #4
steps to solve		 x + 32 = 12
math:  (Note:  since the square is on the number and not on the variable,
the expression qualifies as a linear expression
1. First step is to square the number: x + 32 = 12
x + 9  = 12
2. Second step is to subtract both sides by 9: x + 9 - 9  = 12 - 9
x  = 3
3. Check your work with the original equation: 3 + 32 = 12
 3 + 9  = 12
12  = 12

Flash exercise by Jay Austin and Jordan Noll, College of Design; Brad Hokanson, faculty, College of Design, University of Minnesota, St. Paul, MN; with refinements by: Steve Kladstrup, Independent Flash Developer, Minneapolis, MN.

Thanks to John Hocutt for his contributions to this page – John is a professional math & science tutor based in Redondo Beach, CA

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