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A linear equation in one variable is an equation in which the same letter is used in all variable terms and the
exponent of the variable is 1.
The solution to an equation in one variable is the number that
can be substituted in place of the variable and makes the equation
true. For example 5 is a solution to
the equation 2x + 3 = 13 because 2(5) + 3 = 13 is true.
Equivalent equations are equations that have the same
solutions. For example 2x + 3 = 13
and x = 5 are equivalent equations because each has the
solution of 5.
Basic Principle of Equality
• To preserve equality, if an operation is performed on one side of an equation, the same operation must be performed on the other side.
There are 2 principles (axioms) we will use to solve linear equations in
one variable. The first is the addition principle of equality. This
principle allows us to add (or subtract) the same value to both sides of an equation to obtain an
equivalent equation.
To solve an equation using the addition axiom:
• Locate the variable in the equation.
• Identify the constant that is associated with the variable by addition or subtraction.
• Add the opposite of the constant to both sides of the equation.
Often we need to combine like terms on one or both sides of the
equation before solving.For example:
63 xgives termslike combining
695437 xxx
Whenever variable terms appear on both sides of an equation we use the
addition principle to move all variable terms to the same side, then solve.
For example to solve
8x
xx 685 get tosidesboth 6x to add
When solving these type equations, it makes no difference the side from
which you remove the variable term to start. The goal is to get all variable
terms on one side and all constants on the other.
Whenever quantities appear in parentheses on either side of the
equation they must be removed first.
124 x
1283 xx12243 xx
For each problem so far the coefficient for the variable ended
up being one. We use the multiplication principle (axiom)
to solve equations where the coefficient of the variable is not
one.
The multiplication principle allows us to multiply (or divide) each side of an equation by the same nonzero
quantity to obtain an equivalent equation.
The goal is to get +1 times the variable = a number.
To solve an equation where the coefficient of the variable is not one you need to multiply both
sides of the equation by the reciprocal of the coefficient. An alternate way is to divide both sides by the coefficient of the
variable.
Sometimes it is necessary to combine like terms before
solving the equation.
4538 xx
455 x termslike combine
5by sidesboth divide
9x
When more than one operation is indicated on the variable, undo
addition or subtraction first, then undo multiplication or division next.
Solve: 4x – 2 = 18
• Since the variable has been multiplied by 4 and subtracted by 2, undo by adding 2 and dividing by 4.
4 2 18
4 20
5
4 5 2 18
18
4
1
2
4
8
2x
x
x
Solve: 118 – 22m = 30• Think of 118 – 22m as
118 + ( - 22m)
118 22 30
22 88
22
118 118
2
88
4
118 22 4 30
118 88 30
3
2
30
2
0
2
m
m
m
m
Solve: 5x – 4 = 8x – 13 5 4 5 8 13 5
4 3 13
4 13 3 13 13
9 3
9 3
3 33
x x x x
x
x
x
x
x
5 4 8 8 13 8
3 4 13
3 4 4 13 4
3 9
3 9
3 33
x x x x
x
x
x
x
x
Summary of steps for solving an equation:
• Remove parentheses.• Combine like terms on each side of the
equation.• Sort terms to collect the variable terms on
one side and constants on the other.• Solve for the variable by multiplying by the
reciprocal of the coefficient or dividing by the coefficient of the variable.
5 ( h – 4 ) + 2 = 3h – 4
5h – 20 + 2 = 3h – 4 Distribute.
5h – 18 = 3h – 4
Step 1
Step 2
Combine terms.
5h – 18 + 18 = 3h – 4 + 18 Add 18.
5h = 3h + 14 Combine terms.
Subtract 3h.5h – 3h = 3h + 14 – 3h
2 2
Combine terms.2h = 14
=
h = 7
Step 3 2h 14 Divide by 2.
Solve the following equation.
Check by substituting 7 for h in the original equation.Step 4
5 ( h – 4 ) + 2 = 3h – 4
5 ( 7 – 4 ) + 2 = 3(7) – 4
5 (3) + 2 = 3(7) – 4
15 + 2 = 21 – 4
17 = 17
? Let h = 7.
? Subtract.
True
? Multiply.
The solution to the equation is 7.
Solving an Equation That Has Infinitely Many Solutions
4 ( 2n + 6 ) = 2 ( 3n + 12 ) + 2n
8n + 24 = 6n + 24 + 2n Distribute.
8n + 24 = 8n + 24 Combine terms.
8n + 24 – 24 = 8n + 24 – 24 Subtract 24.
8n = 8n Combine terms.
Subtract 8n.8n – 8n = 8n – 8n
True0 = 0
An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. We
indicate this by writing all real numbers.
Solving an Equation That Has No Solution
6x – ( 4 – 3x ) = 8 + 3 ( 3x – 9 )
6x – 4 + 3x = 8 + 9x – 27 Distribute.
9x – 4 = –19 + 9x Combine terms.
9x – 4 – 9x = –19 + 9x – 9x Subtract 9x.
– 4 = –19 False
Again, the variable has disappeared, but this time a false statement (– 4 = – 19) results. Whenever this happens in solving an
equation, it is a signal that the equation has no solution and we write no solution.
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