Download How to Solve a One-Step Linear Equation

How to Solve a One Step Linear Equation
Example 1
Solve by Adding a Positive Number
a. Solve y – 12 = -21. Then check your solution.
y – 12 = -21
Original equation
y – 12 +12 = -21 + 12
Add 12 to each side.
y = -9
-12 + 12 = 0 and –21 + 12 = -9
To check that –9 is the solution, substitute –9 for y in the original equation.
y – 12 = -21
-9 – 12 = -21
-21 = -21
The solution is –9.
Check
Original equation
Substitute -9 for y.
Subtract.
b. Solve 27 + p = 13. Then check your solution.
27 + p = 13
Original equation
27 + p + (-27) = 13 + (-27)
Add -27 to each side.
p = -14
27 + (-27) = 0 and 13 + (-27) = -14
Check
27 - p = 13
27 - 14 13
13 = 13
The solution is -14.
Original equation
Substitute -14 for p.
Add.
Example 2 Solve by Subtracting
4 1
Solve k + = - . Then check your solution.
5 3
4
1
=3
5
4 4
1 4
k+
- =- 3 5
5 5
17
k=15
k+
4
1
=3
5
17
4
1
+
=15 5
3
1
1
- =3
3
17
The solution is .
15
Check
Example 3
k+
Original equation
4
Subtract 5 from each side.
4 4
1 4 17
5 - 5 = 0 and -3 - 5 = -15 .
Original equation
17
Substitute -15 for k.
Add.
Solve by Adding or Subtracting
Solve m + 4.65 = -3.82 in two ways.
Method 1
Use the Subtraction Property of Equality.
m + 4.65 = -3.82
Original equation
m + 4.65 – 4.65 = -3.82 – 4.65
Subtract 4.65 from each side.
m = -8.47
4.65 – 4.65 = 0 and –3.82 – 4.65 = -8.47
The solution is -8.47.
Method 2
Use the Addition Property of Equality.
m + 4.65 = -3.82
Original equation
m + 4.65 + (-4.65) = -3.82 + (-4.65)
Add -4.65 to each side.
m = -8.47
4.65 + (-4.65) = 0 and -3.82 + (-4.65) = -8.47
The solution is -8.47.
Example 4
Write and Solve an Equation
Write an equation for the problem. Then solve the equation.
The difference of a number and one fourth is negative two thirds.
a number
the difference of
one fourth
is
negative two thirds
n
–
1
4
=
2
-3
1
2
=4
3
1 1
2
1
n– + =- +
4 4
3
4
5
n=12
n–
Original equation
1
Add 4 to each side.
1 1
2 1
5
-4 + 4 = 0 and -3 + 4 = -12
Example 5 Write an Equation to Solve a Problem
Noah received a paycheck on Friday. After buying some athletic shoes for $112, he had $96 left.
How much was his paycheck?
Words
The amount of the paycheck minus the amount of the shoes equals $96.
Variable
Let a = amount of paycheck
Equation The original
amount
minus
the amount
for shoes
equals
–
112
=
a
a – 112 = 96
a – 112 +112 = 96 +112
a = 208
96
Original equation
Add 112 to each side.
-112 + 112 = 0 and 96 + 112 = 208
Noah’s paycheck was for $208.
Example 6
$96
Solve Using Multiplication by a Positive Number
x 1
Solve = - . Then check your solution.
5 6
x
1
=5
6
5
x
=
5
x=-
Original equation
1
5
6
5
6
Multiply each side by 5.
5
x
1
=5
6
5
6 =-1
6
5
1
1
- =6
6
Check
x
5
= x and
1
5
5 = -6 .
6
Original equation
5
Substitute - 6 for x.
5
The solution is - 6 .
Example 7 Solve Using Multiplication by a Fraction
Solve each equation.
7
1
a. 8 r = -14
7
1
r = -1
8
4
7
5
r=8
4
5
8 7
8
r=
4
7 8
7
40
10
r=or 28
7
b. 28 = -7r
28 = -7r
1
1
-7 (28) = -7 (-7r)
-4 = r
Example 8
Original equation
Rewrite the mixed number as an improper fraction.
8
7
Multiply both sides by 7 , the reciprocal of 8 .
Check this result.
Original equation.
1
Multiply each side by -7 , the reciprocal of -7.
Simplify.
Write and Solve an Equation Using Multiplication
One mile is equal to approximately 1.6 kilometers. A local running club is holding a 10-kilometer
race. What is the length of the race in miles?
1.6 kilometers
Words
Variable
time
s
number
of miles
equals
number of
kilometers
in the race
Let m = the number of miles.
Equation
1.6
m
1.6m = 10
=
Original equation
1
1
(1.6m) =
(10)
1.6
1.6
1
Multiply each side by 1.6.
1
1
1.6 (1.6) = 1 and 1.6 (10) = 6.25
m = 6.25
The race is 6.25 miles.
Example 9
Solve Using Division
a. Solve 2.12b = 9.01. Check your solution.
2.12b = 9.01
Original equation
2.12b 9.01
=
2.12 2.12
Divide each side by 2.12.
2.12b
9.01
2.12 = b and 2.12 = 4.25
b = 4.25
Check
2.12b = 9.01
2.12 (4.25) 9.01
9.01 = 9.01
Original equation
Substitute 4.25 for b.
b. Solve –9t = -63.
-9t = -63
Original equation
9t
=
9
Divide each side by –9.
63
9
t=7
Example 10
-9t
-63
= t and -9 = 7
-9
Write and Solve an Equation Using Division
10
Write and solve an equation for the problem below. Then solve the equation.
Eight times a number is negative sixty-four.
eight
times
8
8n = -64
8n
=
8
64
8
n = -8
The solution is –8.
a number
is
negative sixty four
n
=
-64
Original equation
Divide each side by 8.
Check this result.