Solving Equations Solving open sentences using inverse operations.
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Transcript of Solving Equations Solving open sentences using inverse operations.
Solving EquationsSolving EquationsSolving EquationsSolving EquationsSolving open sentences using inverse Solving open sentences using inverse
operations.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 Then, undo the multiplication by dividing by 5.
5 5 x = 3
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 – 4We have two terms with
the variable, 4x and x. 4x + 5 = x - 4We’ll move the variable 4x – x = x – x - 4with the smaller 3x + 5 = -4coefficient, x. To do this we have to look at the sign in front of the variable we’re moving. Since the is nosign 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 = -43x + 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 = 114 + 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 = 122 · 4 · 3 = 12 · 2
24 = 24Division Property of Equality – states you can divide the same amount on both sides of an equation and the equation remains true.
4 · 3 = 124 · 3 = 12
2 212 = 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 = 5Check: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 = 35check:
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 = 2Check:
6x – 2 = 10 6(2) – 2 = 10 Substitute 2 for x
12 – 2 = 10 10 = 10 ? true
Solving equationsGet 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 real-world
problems is the first step to becoming “good” at math.
These skills lay the foundation for studies of algebra, geometry,
and statistics.
Using FormulasUsing FormulasUsing FormulasUsing FormulasFormulas are equations used to show Formulas are equations used to show
relationships between quantities.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 ÷ # of equals miles per
driven gallons gas gallon
m ÷ g = 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).
• Use the formula d = rt to find the indicated variables.1) r = 60 km/h; t = 4 h; d =2) d = 100 km; t = 2 h; r =3) r = 55 km/h; d = 110 km; t =4) r = 35 km/h; t = 3 h; d =5) d = 210 km; t = 7 h; r =6) 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.