Multistep Equations Learning Objectives
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Transcript of Multistep Equations Learning Objectives
Multistep Equations Learning Objectives
Use the properties of equality to solve multistep equations of one
unknown Apply the process of solving multistep equations to solve
multistep inequalities After this lesson, you will be able to use
the properties of equality to solve multistep equations of one
unknown, and apply the process of solving multistep equations to
solve multistep inequalities. How to Identify Multistep
Equations
Some equations can be solved in one or two steps Ex) 4x + 2 = 10 is
a two-step equation Subtract 2 Divide both sides by 4 Ex) 2x 5 = 15
is a two-step equation Add 5 Divide both sides by 2 Some equations
can be solved in one step, and some require two. The equation 4x +
2 = 10 {four x plus two equals ten} is a two-step equation. Solving
it requires first subtracting 2 from both sides and then dividing
both sides by 4. Although this equation requires two steps to
solve, there is only one unknown term (x). The equation 2x 5 = 15
{two x minus five equals fifteen} is another example of a two-step
equation. The variable x appears only once, and solving for this
variable requires only two steps. This image shows a sliding block
puzzle. There are many ways that the blocks can be scrambled, and
solving this type of puzzle requires many steps. However, the
process of solving sliding block puzzles can be learned quickly
because they are always solved the same way. {play animation, read
following sentence as animation begins}. Following a certain
sequence of steps leads to the solution regardless of how the
puzzle was scrambled. How to Identify Multistep Equations
Multistep equation an equation whose solution requires more than
two steps Ex) 5x 4 = 3x + 2 and 4(x 2) = 12 Multistep equations can
take different forms Variables present in two different terms Ex)
6x 2x = 8 + 4,5x 4 = 3x + 2 Ex) 4(x 2) = 12 Like the sliding block
puzzle, some equations require many steps in order to solve. A
multistep equation is an equation whose solution requires more than
two steps. One example is the equation 5x 4 = 3x + 2 {five x minus
four equals three x plus two}. Notice that although there is only
one unknown (x), it is present on both sides of the equation. These
two occurrences of the variable can be combined, but doing this
requires extra steps. The equation 4(x 2) = 12 {four times x minus
two equals twelve} is another example of a multistep equation. In
this example, the distributive property must be used in order to
simplify the left side of the equation. As a result, this equation
requires more than two steps to solve. Although multistep equations
can take different forms, there are certain features that indicate
when an equation requires multiple steps in order to solve. One is
if the variable is present in two different terms. These terms
could be on the same side of the equation, as in 6x 2x = {six x
minus two x equals eight plus four}, but frequently they are on
opposite sides, as in 5x 4 = 3x + 2 {five x minus four equals two x
plus two}. Another feature that can indicate a multistep equation
is the presence of parentheses on either side of it, as in 4(x 2) =
12 {four times x minus two equals twelve. Some characteristics that
indicate that an equation may require more than two steps are
listed. Characteristic Example Multiple variable or constant terms
on the same side x + 2x + 3x 1 = Variable present on both sides 4x
2 = 3x + 3 Parentheses present on either side 3(x + 2) = 21
Combining Terms First step to solving a multistep equation is to
simplify each side Use the distributive property to eliminate
parentheses Ex) 5(x 2) + x = 2(x + 3) + 4 simplifies to 5x 10 + x =
2x Combine like terms Ex) 5x 10 + x = 2x Combine 5x and x to 6x on
the left side Combine 6 and 4 to 10 on the right side Simplifies to
6x 10 = 2x + 10 Terms containing variables cannot be combined with
constant terms The first step in solving a multistep equation is to
simplify each side. This means using the distributive property, if
necessary, to eliminate parentheses. Both sides of the equation 5(x
2) + x = 2(x + 3) + 4 {five times x minus two plus x equals two
times x plus three plus four} contain parentheses that can be
eliminated in this way. The distributive property simplifies this
equation to 5x 10 + x = 2x {five x minus ten plus x equals two x
plus six plus four}. Simplifying each side also involves combining
like terms. On the left, the terms 5x and x can be combined to
6xsince each is a multiple of the same variable. On the right, the
constant terms 6 and 4 can be combined. This produces the
simplified equation 6x 10 = 2x + 10 {six x minus ten equals two x
plus ten}. It is important to remember that terms containing the
variable cannot be combined with constant terms. Combining Terms
Consider 5(x 3) + 4 = 3(x + 1) 2
Distributive Property can be used on both sides due to the
parentheses multiplied by constants Produces 5x = 3x + 3 2 Terms
can be combined on both sides Combine 15 and 4 on the left and 3
and 2 on the right Produces 5x 11 = 3x + 1 Another example of a
problem that requires combining like terms is the equation 5(x 3) +
4 = 3(x + 1) 2 {five times x minus three plus four equals three x
plus one minus two}. The numbers multiplied by expressions in
parentheses make it clear that the distributive property must be
used on both sides of this equation. Using the distributive
property on each side produces 5x = 3x + 3 2 {five x minus fifteen
plus four equals three x plus three minus two}. This new equation
has terms on each side that can be combined, the 15 and 4 on the
left and the 3 and 2 on the right. This simplifies the equation to
5x 11 = 3x + 1 {five x minus eleven equals three x plus one}. How
to Solve Multistep Equations
Consider the equation 4(x + 2) 10 = 2(x + 4) Simplify with the
distributive property 4x + 8 10 = 2x + 8 Combine like terms 4x 2 =
2x + 8 Eliminate the unknown from one side 2x 2 = 8 Eliminate the
constant term on the other side 2x = 10 Divide each side by the
coefficient of the variable x = 5 Multistep equations can usually
be solved by a straightforward process. The first step is to
simplify each side by eliminating grouping symbols and combining
like terms. Consider the equation 4(x + 2) 10 = 2(x + 4) {four
times x plus two minus ten equals two times x plus four}. The
presence of parentheses, as well as the apparent complexity of the
equation, indicates that this is a multistep equation. Each side
can be simplified by the distributive property, producing the form
4x + 8 10 = 2x + 8 {four x plus eight minus ten equals two x plus
eight}. The left side can now be simplified further because it
contains two constant terms. Combining these simplifies the
equation to 4x 2 = 2x + 8 {four x minus two equals two x plus
eight}.The equation is now simpler in form than the original, but
the unknown (x) still appears on both sides. It is easier to solve
for the unknown if it occurs only once in the equation. As a
result, the second step of solving a multistep equation is to
eliminate the unknown from one side of the equation. It can be
eliminated from either side, but since the variable is
traditionally written on the left, it is more common to eliminate
it from the right side of the equation. In this example,
subtracting 2x from each side of the equation eliminates the
variable from the right side, leaving 2x 2 = 8. This is now a
two-step equation since the variable occurs only once, and only two
operations are performed on it, namely multiplication and
subtraction. As a result, it can now be solved as a two-step
equation by undoing each of these operations. The next step is to
eliminate the constant term on the side with the variable. Adding 2
to both sides yields 2x = 10. The final step is to divide both
sides by the coefficient of the variable. Dividing each side of
this equation by 2 results in the solution, x = 5. Steps to Solve a
Multistep Equation
How to Solve Multistep Equations Same general set of steps is
useful in solving many multistep equations Steps to Solve a
Multistep Equation First step Simplify each side as much as
possible Second step Eliminate the variable from one side Third
step Eliminate the constant term from the side with the variable
Fourth step Divide each side by the coefficient of the variable
Solving this equation required only four basic steps. Although
multistep equations can appear in many different forms, the same
general set of steps is useful in solving many of them. After
identifying the problem as a multistep equation, apply the
following steps. First, simplify each side of the equation as much
as possible by combining like terms and using the distributive
property if necessary in order to remove parentheses. Second,
eliminate the variable on one side of the equation. Third,
eliminate the constant term on the side with the variable. Fourth,
divide each side by the coefficient of the variable. Some multistep
equations may not require every step. For example, if the variable
has no coefficient after the second step is completed, then the
fourth step is unnecessary. How to Solve Multistep Equations
Example
Ex) Mark and Susan are given the same amount of money. Mark spends
$5, and Susan spends $20. If Mark now has twice as much money as
Susan, how many dollars did they each have originally? Analyze Find
the original amounts Formulate Represent the problem as an equation
Determine x 5 = 2(x 20) x 5 = 2x 40 x 5 = 40 x = 35 x = 35 Justify
They each began with $35 Evaluate 35 5 is double 35 20 Mark and
Susan are given the same amount of money. Mark spends $5, and Susan
spends $20. If Mark now has twice as much money as Susan, how many
dollars did they each have originally? First, analyze the problem.
The problem presents a word problem about Mark and Susan, who begin
with equal amounts of money. When Mark spends $5 and Susan spends
$20, Mark has twice as much as Susan. The problem asks for the
original amount that they each had. Next, formulate a plan or
strategy to solve the problem. Represent the problem as an
equation. Simplify each side, eliminate the variable from one side
and the constant from the other side, and divide both sides by the
coefficient of the variable. Next, determine the solution to the
problem. Represent the problem as an equation. Use the distributive
property. Subtract 2x from each side. Add 5 to each side. Divide
each side by 1. Now that the solution has been determined, justify
it. An equation was written showing Marks amount (on the left)
equal to twice Susans amount (on the right). The right side of the
equation was simplified, and then the variable term was eliminated
from the right, and the constant term was eliminated from the left.
Dividing each side by the coefficient of x gave the solution, which
shows that Mark and Susan each began with $35. Last, evaluate the
effectiveness of the steps, and the reasonableness of the solution.
The steps for solving a multistep equation worked effectively. The
answer is reasonable because 35 5is double 35 20. Consecutive
Integers Consecutive integers integers that are separated by
exactly one unit Ex) 5 and 6 When domain is limited to integers,
the letter n is used to represent the variable Ex) Find 3 integers
that sum to 24 Solution is 7, 8, 9 Set of consecutive integers from
1 to 3 Consecutive integers are integers that are separated by
exactly one unit. The numbers 5 and 6 are consecutive integers
because there is no other integer in between them. Here is a set of
consecutive integers. Note that each number in this set is equal to
the previous number plus 1. When the domain of a variable is
limited to integer values, the letter n is frequently used to
represent that variable. Problems involving consecutive integers
usually can be expressed as multistep equations. In these
equations, the integer with the lowest value is usually written as
n because this letter is generally used for variables that are
restricted to integer values. Since the first integer is n, the
second must be n + 1, the third being n + 2, and so forth. For
example, suppose that a problem asks for three consecutive integers
whose sum is 24. Since the three integers n, n +1, and n + 2 are
added together, this problem can be written as the equation n + n n
+ 2 = 24 {n plus n plus one plus n plus two equals twenty four}.
This equation can be solved by following the sequence of steps for
solving multistep equations. The three integers in this situation
would be 7, 8, and 9. Consecutive Integers Ex) Find three
consecutive integers where the sum of the first two is equal to 3
times the third one Can be represented as n + (n + 1) = 3(n + 2)
Can be solved using the process to solve multistep equations As
another example, suppose a problem asks for three consecutive
integers where the sum of the first two is equal to 3 times the
third one. This problem can be represented by the equationn + (n +
1) = 3(n + 2) {n plus n plus one equals three times n plus two}.
Again, this equation can be solved by following the sequence of
steps for solving multistep equations. Consecutive Integers
Consecutive even integers are spaced two units apart Ex) 6, 8, and
10 Can be represented as n, n + 2, n + 4 Consecutive odd integers
represented the same way Ex) 7, 9, and 11 Can be represented as n,
n + 2, n + 4 Check answer to make sure the solutions are odd Some
problems ask for consecutive even integers or consecutive odd
integers. Consecutive even integers are not one unit apart because
any even number plus 1 is an odd number. As a result, consecutive
even integers are spaced two units apart. The numbers 6, 8, and 10
are consecutive even integers. Any set of three consecutive even
integers can be represented as n, n + 2, and n + 4. Consecutive odd
integers, such as 7, 9, and 11, can also be represented as n, n +
2, and n + 4 because they are also spaced two units apart. Even and
odd integers are represented the same way in an equation. As a
result, if a problem asks for a set of consecutive odd integers, it
is important to check the answer at the end to make sure that the
solutions obtained are actually odd. RATIOS, RATES, &
PROPORTIONS RATIOS A ratio is the comparison of two quantities with
the same unit.
A ratio can be written in three ways: As a quotient (fraction in
simplest form) As two numbers separated by a colon (:) As two
numbers separated by the word to Note:ratios are unitless (no
units) Ex: Write the ratio of 25 miles to 40 miles in simplest
form.
What are we comparing? miles 25 miles to 40 miles Units, like
factors, simplify (divide common units out) Simplify The ratio is
5/8 or 5:8 or 5 to 8. RATES A rate is the comparison of two
quantities with different units.
A rate is written as a quotient (fraction) in simplest form.
Note:rates have units. Ex: Write the rate of 25 yards to 30 seconds
in simplest form.
What are we comparing? yards & seconds 25 yards to 30 seconds
Units cant simplify since they are different. Simplify The rate is
5 yards/6 seconds. UNIT RATES A unit rate is a rate in which the
denominator number is 1.
The 1 in the denominator is dropped and often the word per is used
to make the comparison. Ex:miles per hour mph miles per gallon mpg
Ex: Write as a unit rate 20 patients in 5 rooms
What are we comparing? patients & rooms 20 patients in 5 rooms
Units cant simplify since they are different. Simplify The rate is
4 patients/1room Four patients per room PROPORTIONS A proportion is
the equality of two ratios or rates.
If a/b and c/d are equal ratios or rates, then a/b = c/d is a
proportion. In any true proportion the cross products are equal:
Why? Multiply thru by the LCM Simplify (bd) (bd) ad = bc Cross
products are equal! We will use the property that the cross
products are equal for true proportions to solve proportions. x 6
Ex:Solve the proportion x 6 72 If the proportion is to be true, the
cross products must be equal find the cross product equation: 7x =
(12)(42) 7x = 504 x = 72 Ex: Solve the proportion
If the proportion is to be true, the cross products must be equal
find the cross product equation: 24 = 3(n 2) 24 = 3n 6 30 = 3n x 2
10 = n Check: x 2 Ex:Solve the proportion
If the proportion is to be true, the cross products must be equal
find the cross product equation: (5)(3) = 7(n + 1) 15 = 7n + 7 8 =
7n 8/7 = n Check: