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Lesson 2 – 4: Solving Equations with Variables on Both Sides
2-4.1 - Solving equations with variables on both sides
To solve equation that have variables on both sides we must collect the variables terms so that they are on only
one side. We do this be adding or subtracting the variables to make one side zero. This is step 3 of the 5 for
solving equations
5 Steps to solve Equations:





Distribute to get rid of Parentheses
Combine like Terms
Get Variables to one side
Inverse of Addition and Subtraction
Inverse of Multiplication and Division
Examples: Solve each equation
5 x  2  3x  4
7 k  4k  15
4k  4k
3k  15
k 5
3 x
 3x
2x  2  4
2
2x  6
2
x3
2-4.2 - Simplifying both sides before solving
Examples: Solve each equation
2( y  6)  3 y
2 y  12  3 y
2 y
 2y
12  y
3  5 x  2 x  2  2(1  x)
3  5 x  2 x  2  2  2 x
3  3x  4  2 x
 3x
 3x
3  4  5 x
4  4
7  5x
7
x
5
2-4.3 - Equations with infinitely many solutions or no solutions
Vocabulary:
Identity - an equation that is true for all values of the variable. It has infinite solutions
Contradiction - an equation that is not true for any value of the variable. It has no solutions
Sometimes equations are true no matter what value is substituted in for the variable. In this case, the equation is
called an identity and is said to have infinitely many solutions.
On the other hand, sometimes no matter what value is put into an equation for the variable nothing will work
and it has no solutions. This is said to be a contradiction.
Examples: Solve each equation
x  4  6 x  6  5x  2
5 x  4  4  5 x
5 x
 5x
44
Infinite number of solutions
8 x  6  9 x  17  x
x  6  17  x
x
x
6  17
No solutions