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Section 2 – 1:
Solving 1 and 2 Step Equations
Expressions
The last chapter in this book contained expressions. The next type of algebraic statement that we will
examine is an equation. At the start of this section we need to clarify the difference between an
expression and an equation.
Equations contain an equal sign = and expressions do not contain an equal sign
Examples of Expressions:
2(4 − 7)2
3(5) − 4
6x 2 − 2x 2
5x + 4 − 3x + 8
You have learned to do two things with expressions so far:
1. Simplify an expression
to a more simple expression.
3(5) − 4
simplifies to
15 − 4
simplifies to
11
2. Evaluate the value of an expression
when the value for the variable is given
(4 − 7)2
simplifies to
3x + 5
where x = 2
simplifies to
3(2) + 5
simplifies to
11
(−3)2
simplifies to
9
x 2 − 3x
where x = 5
simplifies to
(5)2 − 3(5)
simplifies to
10
.
Notice that we DO NOT use equal signs when we simplify expressions.
We work down the page using the PEMDAS steps.
Each step downward changes the first expression into a simpler expression.
The final answer is still an expression but it is in a simpler form then the first expression.
Equations
Equations are made up of two expressions separated by an equal symbol.
6=6
5+ 4 =9
5 − 7 = −2
The equations above are always true because the equations contain only numbers and the left
and right sides of the equation are equal to each other.
x +3= 8
4x = 8
4x = 8
The equations above cannot be determined to be true or false because the value for x is not given so
we do not know if the left and right sides of the equation are equal to each other. We will
now learn how to find the value for x that will make the left and right sides of the equation are
equal to each other. We call this process finding a solution to an equation. We also say that
we are solving the equation.
Math 100 Section 2 – 1
Page 1
© 2015 Eitel
The Solution to an Equation
The solution to an equation is an equation with x all alone on one side of the = symbol and a
number on the other side. You read the solution by reading the variable side first then the
value for the variable.
Examples of solutions to an equation
READ x FIRST no matter which side of the equal sign it is on.
x =5
is read
x is equal to 5
7=x
is read
x is equal to 7
−3 = x
is read
x is equal to − 3
Determine if a value of x is a solution to an equation
If a given value for x is a solution to an equation
that value of x can be substituted into the equation for the x variable
and the resulting equation will be TRUE.
If the resulting equation is FALSE then that number is not a solution.
Substitute the given value of x for the variable and determine if it is a solution.
Example 1
Example 2
Is x = −3
a solution to the equation
−4 x + 1= 13
Is 5 = x
a solution to the equation
19 = 2x + 8
replace x with − 3 in
−4 x + 1= 13
replace x with 5 in
19 = 2x + 8
this yeilds
−4(−3) + 1 = 13
which reduces to
13 = 13
this yeilds
19 = 2(5) + 8
which reduces to
19 = 18
so x = −3
IS A SOLUTION to the equation
−4 x + 1= 13
so 5 = x
IS NOT A SOLUTION to the equation
19 = 2x + 8
Math 100 Section 2 – 1
Page 2
© 2015 Eitel
Example 3
Example 4
Is x = 4
a solution to the equation
2x + 1= 4 x − 7
Is x = −1
a solution to the equation
−3x + 1 = 2x + 3
replace each x with 4 in
2x + 1= 4 x − 7
replace each x with − 1 in
−3x + 1 = 2x + 3
this yeilds
2(4) + 1= 4(4) − 7
which reduces to
8 + 2 = 16 − 7
which reduces to
9=9
this yeilds
−3(−1) + 1 = 2(−1) + 3
which reduces to
3 + 1= −2 + 3
which reduces to
4=1
so x = 4
IS A SOLUTION to the equation
−2 x + 1 = 4x − 7
so x = −1
IS NOT A SOLUTION to the equation
−3x + 1 = 2x + 3
Math 100 Section 2 – 1
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© 2015 Eitel
Solving for x: Finding the Solution to an Equation
Solving an equation consists of performing a series of operations to the original equation to get x all
alone on one side of the = symbol and a number on the other side. Each operation will create an
equivalent equation. Equivalent equations have the same solution as the original equation but are in
a more simple form. The operations will eliminate the constants that have been added or subtracted
from the variable term and then eliminate the coefficients that have been multiplied by or divided into
the variable.
The following four properties state the four operations that can be performed on any equation to
produce an equivalent equation.
The Addition Property
of Equality
The Subtraction Property
of Equality
You can add the same number
to both sides of an equation
and still have an equivalent equation
You can subtract the same number
from both sides of an equation
and still have an equivalent equation
If x − 3 = 10
you add 3 to both sides
of x − 3 = 10
+3 +3
to get the solution
x = 13
Check: 13 − 3 = 10
If 4 = x + 9
you subtract 9 from both sides
of 4 = x + 9
−9
−9
to get the solution
−5 = x
Check: 4 = −5 + 9
The Multiplication Property
of Equality
The Division Property
of Equality
You can multiply both sides of an
equation by the same number
and still have an equivalent equation
You can divide both sides
of an equation by the same number
and still have an equivalent equation
x
−3
you multiply both sides by − 3
x
(−3)5 = (−3)
−3
to get the solution
−15 = x
−15
Check: 5 =
−3
Note: Parenthesis ( ) must be used
to show multiplication.
If 21 = −7 x
you divide both sides by − 7
21 −7 x
=
−7 −7
to get the solution
−3 = x
Check: 21 = −7(−3)
If 5 =
Math 100 Section 2 – 1
Note: A fraction bar must be used
to show division.
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© 2015 Eitel
Solving 1 Step Equations
One Step Equations are equations that require the use of one of the four properties of equality.
To solve for x (get x alone):
Eliminate the constant term: Add or Subtract
Example 1
If a constant has been subtracted from the variable term
then add the constant to both sides of the equation
and simplify both sides of the equation
Example 2
Solve for x
x− 4= 2
Solve for x
x− 5= −7
add 4 to both sides of the equation
add 5 to both sides of the equation
x− 4= 2
+ 4 +4
the left side simplifies to x
the right side simplifies to 6
x =6
x− 5= −7
+5 +5
the left side simplifies to x
the right side simplifies to − 2
x = −2
check: 6 − 4 = 2
check: − 2 − 5 = −7
If a constant has been added to the variable term
then subtract the constant from both sides of the equation
and simplify both side of the equation.
Example 3
Example 4
Solve for x
x +3 = 8
Solve for x
x + 10 = − 4
subtract 3 from both sides of the equation
subtract 10 from both sides of the equation
x+ 3= 8
−3 −3
the left side simplifies to x
the right side simplifies to 5
x =5
x + 10 = − 4
− 10 − 10
the left side simplifies to x
the right side simplifies to − 14
x = −14
check: 5 + 3 = 8
check: − 14 + 10 = −4
Math 100 Section 2 – 1
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© 2015 Eitel
Eliminate the coefficient : Multiply or Divide
If the variable has been multiplied by a number then
divide both sides of the equation by that number
and simplify both sides of the equation
Example 5
Example 6
Solve for x
3x = 12
Solve for x
− x= − 6
divide both sides of the equation by 3
divide both sides of the equation by − 1
3/ x 12
=
3/
3
the left side simplifies to x
the right side simplifies to 4
−/ 1/ x −6
=
−/ 1/ −1
the left side simplifies to x
the right side simplifies to 6
x= 6
x= 4
check: − (6) = − 6
check: 3(4) = 12
If the variable has been divided by a number
then multiply both sides of the equation by that number
and simplify both sides of the equation.
Example 7
Example 8
Solve for x
x
= 5
4
Solve for x
x
= 2
−3
multiply both sides of the equation by 4
multiply both sides of the equation by − 3
x
= 5(4)
4/
the left side simplifies to x
the right side simplifies to 20
x
= 2(−3)
− 3/
the left side simplifies to x
the right side simplifies to − 6
(− 3/ )
(/4)
x = 20
check:
x = –6
20
=5
4
Math 100 Section 2 – 1
check:
Page 6
−6
=2
−3
© 2015 Eitel
Solving 2 Step Equations
Two Step Equations are equations that require the use of two of the four properties of equality to
find a solution for the equation.
To solve for x (get x alone):
Step One: Eliminate the constant term: Add or Subtract
If the constant has been added to the variable term, subtract it from both sides of the equation.
If the constant has been subtracted from the variable term then add it to both sides of the equation.
You should now have a variable with a coefficient on one side of the equal sign
and a constant on the other side of the equal sign.
Step Two: Eliminate the coefficient. Multiply or divide.
If the variable has been multiplied by a number then
divide both sides of the equation by that number.
If the variable has been divided by a number then
multiply both sides of the equation by that number.
Example 9
Math 100 Section 2 – 1
Example 10
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© 2015 Eitel
Solve for x
3x − 4 = 8
Solve for x
22 = 2x + 4
add 4 to
both sides of the equation
subtract 4 from
both sides of the equation
3x − 4 = 8
+4 +4
22 = 2x + 4
−4
−4
3x = 12
18 = 2x
divide both sides of the
equation by 3
divide both sides of the
equation by 2
3x 12
=
3
3
18 2x
=
2
2
x= 4
check: 3(4) − 4 = 8
Example 11
Math 100 Section 2 – 1
9= x
check: 22 = 2(9) + 4
Example 12
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© 2015 Eitel
Solve for x
x
− 5 = −1
2
Solve for x
x
+1= 5
−3
add 5 to
both sides of the equation
subtract 1 from
both sides of the equation
x
− 5 = −1
2
+5 +5
x
+1 = 5
−3
−1 − 1
x
= 4
2
x
= 4
−3
multiply both sides of the equation by − 3
multiply both sides of the equation by 2
(2/ )
x
= 4(2)
2/
(− 3/ )
x=8
check:
x = – 12
8
− 5 = −1
2
check:
4 − 5 = −1
−1 = −1
Math 100 Section 2 – 1
x
= 4(−3)
− 3/
−12
+1 =5
−3
4 +1 = 5
5=5
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© 2015 Eitel