Download Solving Equations

Solving Equations
Solving open sentences using inverse
operations.
What will happen if you
add or subtract an equal
amount of weight on
both sides of the
scales?
Solving equations is like
balancing scales, we
must always keep the
sides equal.
Solving equations is just a matter of undoing
operations that are being done to the variable.
In a simple equation, this may mean that we
only have to undo one operation, as in the
following example.
Solve the following equation for x
x+3=8
x+3=8
the variable is x
x + 3 – 3 = 8 – 3 we are adding 3 to the variable, so to get
rid of the added 3, we do the opposite--subtract 3.
x=5
remember to do this to both sides of the
equation.
In an equation which has more than one operation,
we have to undo the operations in the correct
order. We start with the operation the farthest
away from the variable.
Solve the following equation: 5x – 2 =13
5x – 2 = 13
The variable is x
5x – 2 + 2 = 13 + 2 We are multiplying it by 5, and subtracting 2
First, undo the subtracting by adding 2.
5x = 15
5 5
x=3
Then, undo the multiplication by dividing by 5.
Suppose there are variables on both sides of the
equation. The trick now, is to get the variables on
the same side by adding or subtracting them.
Solve for x in the equation 4x + 5 = x – 4
We have two terms with
the variable, 4x and x.
We’ll move the variable
4x + 5 = x - 4
4x – x = x – x - 4
3x + 5 = -4
with the smaller
coefficient, x. To do this
we have to look at the
sign in front of the variable
we’re moving. Since the is no
sign we know it is +. To move this
Variable we do the opposite, so we’’ll subtract
x from both sides.
Now we proceed as before:
3x + 5 = -4
3x + 5 – 5 = -4 – 5 Subtract 5 from both sides.
3x = -9
3
3
Divide both sides by 3.
x = -3
With any math there are new
vocabulary words and rules we must
follow. Let’s look at some of the new
terms and rules before we move on.
Solving Equations by Adding or Subtracting
Equation – a mathematical sentence that shows two
expressions are equal.
Solve – to find the answer or solution.
Solution – the value that makes an equation true.
Inverse operations – operations that “undo” each
other; addition and subtraction, multiplication and
division.
Isolate the variable – to get the variable on one
side of an equation or inequality by itself in order
to solve.
Open sentence – an equation that contains at least
one variable.
Addition Property of Equality – states you
can add the same amount to both sides of an
equation and the equation remains true.
2+3=5
2+3+4=5+4
9 = 9 ? true
Subtraction Property of Equality – states you
can subtract the same amount from both sides
of an equation and the equation remains true.
4 + 7 = 11
4 + 7 – 3 = 11 – 3
8 = 8 ? true
Addition and subtraction are inverse operations,
which means they “undo” each other. To solve an
equation, use inverse operations to isolate the
variable, or to get the variable on one side of the
equal sign by itself.
x+4=9
subtract 4 from both sides
x + 4 – 4 = 9 – 4 Subtraction property of equality
x + 0 = 5 Identity Property of Zero: x + 0 = 5
check:
x+4=9
5 + 4 = 9 substitute 5 for x
9 = 9 ? true
w – 3 = 9 Add 3 to both sides
w – 3 + 3 = 9 + 3 Addition Property of Equality
w + 0 = 12 Identity Property of Zero: w + 0 = w
check:
12 – 3 = 9 Substitute 12 for w
9 = 9 ? True
It is very important to write all the steps and check
your answer each time you solve an equation.
Solving Equations by Multiplication or Division
Multiplication Property of Equality – states you can
multiply the same amount on both sides of an
equation and the equation remains true.
4 · 3 = 12
2 · 4 · 3 = 12 · 2
24 = 24
Division Property of Equality – states you can divide
the same amount on both sides of an equation and
the equation remains true.
4 · 3 = 12
4 · 3 = 12
2
2
12 = 6
2
Multiplication and Division are inverse
operations, which means they “undo” each
other. To solve an equation, use inverse
operations to isolate the variable, or get the
variable on one side of the equal sign by itself.
7x = 35 Divide both sides by 7.
7x = 35 Division Property of Equality
7
7
1x = 5
1·x=x
X=5
Check:
7x = 35
7 (5) = 35 substitute 5 for x
35 = 35 ? true
n ÷ 5 = 7 Multiply both sides by 5
n ÷ 5 · 5 = 7 · 5 Multiplication Property of Equality
n = 35
check:
n÷5=7
35 ÷ 5 = 7 Substitute 35 for n
7 = 7 ? True
It is very important to write all the steps and check
your solution each time you solve an equation.
Sometimes it is necessary to solve equations
by using 2 or more inverse operations. For
instance, the equation 6x – 2 = 10.
Always start with the operation that is the
farthest away from the variable.
6x – 2 = 10 Add 2 to both sides first.
6x – 2 + 2 = 10 + 2 Addition Property of Equality
6x = 12 Divide both sides by 6
6 6 Division Property of Equality
x=2
Check:
6x – 2 = 10
6(2) – 2 = 10 Substitute 2 for x
12 – 2 = 10
10 = 10 ? true
Solving equations
Get you pencil and calculator ready and try
these problems.
1) m + 15 = 25
2) 50 = h – 3
3) 4d = 144
4) x/3 = 18
5) S + 2 = 13
6) 4x + 3 =19
7) y/2 – 5 = 1
8) 26 = 3f + 10f
9) 4(2x -1) + 3x = 11
10) 144 = 12h
Evaluating and solving
simple expressions and
equations, using order of
operations, and using
variables to solve realworld problems is the
first step to becoming
“good” at math. These
skills lay the foundation
for studies of algebra,
geometry, and statistics.
Using Formulas
Formulas are equations used to show
relationships between quantities.
Using Formulas (equations)
A formula or equation shows the relationship among
certain quantities. The formula below can be used
to find the miles per gallon achieved by a car.
number of miles
driven
÷
# of
gallons gas
m
÷
g
equals
=
miles per
gallon
mpg
You drove 294 miles before stopping to get gas.
Your gas tank holds 12 gallons of gas. What gas
mileage does your car get?
294 ÷ 12 = 24.5 mpg
The formula was distance traveled by a moving object is
d = rt, where d represents distance in kilometers (km),
r represents the rate in kilometers per hour (km/h),
t represents the time in hours (h).
•
1)
2)
3)
4)
5)
6)
Use the formula d = rt to find the indicated variables.
r = 60 km/h; t = 4 h; d =
d = 100 km; t = 2 h; r =
r = 55 km/h; d = 110 km; t =
r = 35 km/h; t = 3 h; d =
d = 210 km; t = 7 h; r =
r = 80 km/h; d = 320 km; t =
The formula I = prt is used to find the amount of
simple interest on a given amount, where I is the
interest; p is the principal amount; r is the rate of
percent; and t is the time in years. Thurman borrowed
$13,500 from his brother for 4 years at an annual
percentage rate of 6%. How much interest will he pay
if he pays the entire loan off at the end of the fourth
year? What is the total amount he will repay?
Formulas are used
everyday to solve problems,
whether you are computing
gas mileage for your car
(mpg = m ÷ g) or changing
degrees Celsius to
Fahrenheit (F = 9/5C + 32),
or even solving the
Pythagorean Theorem
(a² + b² = c²) to find
distance.