Download Linear equations with fractional terms (rational equations). As any

Linear equations with fractional terms (rational equations).
As any linear equation, rational equations should be solved starting by getting the variable in one side of the equation
(if that is the case), and then start isolating it (getting it by itself).
The easiest way to solve these equations is to get only one fraction in each side of the equation, and then crossmultiply.
Rational equation
Description of the procedure
𝑦 1 2𝑦 1
+ =
−
2 6
3 3
𝑦 2𝑦
1 1
−
=− −
2 3
6 3
To start you will get the variables in one side of the equation,
2𝑦
and the constants in the other side by subtracting 3 to both
𝑦x3 2𝑦x2
1 1 x2
−
=− −
2x3 3 x2
6 3 x2
3𝑦 4𝑦
1 2
−
=− −
6
6
6 6
3𝑦 − 4𝑦
−1 − 2
(
)∙6=(
)∙6
6
6
1
6
sides, and subtracting to both sides of the equation.
Now, to simplify the equation you get the same denominator
for all the fractions. If you multiply by this common
denominator each one of the fractions you can cancel them
out. Simplify the resulting equation.
−𝑦 = −3
𝑦=3
Here is the final answer.
Linear inequalities.
Linear inequalities, the same as linear equations, should be solved starting by getting the variable in one side of the
inequality (if that is the case), and then start isolating it (getting it by itself). The only difference between solving linear
equations and linear inequalities is when the inequalities are multiplied or divide by negative numbers, in those cases
the inequality symbol has to be flipped.
Inequality
Description of the procedure
2(𝑡 + 2) − 4𝑡 ≥ −2
To start solving the inequality you need to use the distributive
property.
Now you combine like terms.
2𝑡 + 4 − 4𝑡 ≥ −2
4 − 2𝑡 ≥ −2
4 − 4 − 2𝑡 ≥ −2 − 4
−2𝑡 ≥ −6
−2𝑡 −6
≥
−2
−2
𝑡≤3
Now you subtract 4 to both sides of the inequality.
Now you divide by −2. (remember to flip the sign of the
inequality)
Here is the final answer.
Equations of a line.
To find the x-intercept of a line plug-in in its equation a value of zero for “y” and solve for “x”. in a similar way, to find
the y-intercept of a line plug-in in its equation a value of zero for “x” and solve for “y”. The slope of a line can be found
𝑦 −𝑦
with the formula 𝑚 = 𝑥2 −𝑥1 . To sketch graph of a line using the intercepts, plot the coordinates of them ((0,y) for the
2
1
y-intercept, and (x,0) for the x-intercept), and then sketch the line using these two points.
Literal equations.
Literal equations are equations with only variables and constants. Solving a literal equation for a given variable consists
on getting that given variable by itself using mathematical operations.
Literal equation
Description of the procedure
1
𝑉 = 𝜋𝑟 2 ℎ; 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 "𝑟"
3
1
3𝑉 = ( 𝜋𝑟 2 ℎ) ∙ 3
3
3𝑉 = 𝜋𝑟 2 ℎ
3𝑉 𝜋𝑟 2 ℎ
=
𝜋ℎ
𝜋ℎ
3𝑉
= 𝑟2
𝜋ℎ
√
3𝑉
= √𝑟 2
𝜋ℎ
First you multiply by 3 both sides of the equation to remove
the fraction on the right side of the equation.
Now you divide both sides of the equation by “πh”
Now you square root both sides of the equation to get our
final answer.
3𝑉
√ =𝑟
𝜋ℎ
Rules of exponents.
Rules
Description of the rules
 To multiply like terms multiply the coefficients, if any, and
add the exponents of the like variables.
 To divide like terms divide the coefficients, if any, and
subtract the exponents of the like variables.
 When you have entire expressions raised to a given
exponent, raise the coefficients, if any, to the given
exponent, and for variables multiply the exponents.
 Don’t forget to turn any negative exponent into positive.
Simplifying rational expressions.
Rational expression
Description of the procedure
First you need to factor the numerator and the denominator.
In this case the numerator can be factored using case #1 of
factoring, and the denominator by factoring out “3”
At this point any common factors in the numerator and the
denominator can be simplified, but before you do so you
need to state the restriction on the variable – values you
cannot use for the variable because they would make the
expressions undefined (divided by zero) –
Here is your final answer.
Multiplication of rational expressions.
Rational expression
4𝑥(𝑥 − 1)
(𝑥 + 3)(𝑥 − 2)
∙
(𝑥 + 3)(𝑥 − 1)
4𝑥
Description of the procedure
First you need to factor the numerator and the denominator
of both rational expressions. In this case, in the first fraction,
the numerator can be factored by factoring out “4x”, and the
denominator can be factored by using case #1 of factoring. In
the second fraction the numerator can be factored by using
case #1 of factoring
At this point any common factors in the numerator and the
denominator can be simplified, or you could multiply straight
across.
By multiplying straight across you would get the expression
on the left.
You can now simplify the common factors, but before you do
so you need to state the restriction on the variable – values
you cannot use for the variable because they would make the
expressions undefined (divided by zero) –
Here is your final answer.
Division of rational expressions.
Rational expression
2𝑥
(𝑥 − 2)(𝑥 − 4)
∙
3(𝑥 − 4)
𝑥(𝑥 − 2)
Description of the procedure
First you need to use what is called “keep, change, flip” to
turn the division into a multiplication, and then solve it as
one.
Now you can factor the numerator and the denominator of
both expressions. In this case, in the first fraction, the
denominator can be factored by factoring out “3”. In the
second fraction the numerator can be factored by using case
#1 of factoring, and the denominator can be factored by
factoring out “x”.
At this point any common factors in the numerator and the
denominator can be simplified, or you could multiply straight
across.
By multiplying straight across you would get the expression
on the left.
You can now simplify the common factors, but before you do
so you need to state the restriction on the variable – values
you cannot use for the variable because they would make the
expressions undefined (divided by zero) –
Here is your final answer.
Addition and subtraction of rational expressions.
Rational expression
−2𝑥 + 21
; 𝑥 ≠ 3, 𝑥 ≠ −2
(𝑥 − 3)(𝑥 + 2)
Description of the procedure
To add or subtract fractions you need to have a common
denominator. In cases like this one you have to multiply each
fraction by the denominator of the other one to get the
common denominator you need.
Now that you have the common denominator you can
add/subtract the rational expressions the same way you
would do with fractions with like denominators: Keep the
common denominator and add/subtract the numerators.
At this point, after distributing in the numerators, you
combine like terms.
Here is your final answer after stating the restriction for the
variable.
Simplifying radical expressions.
Radical expression
√36 ∙ 2 ∙ 𝑥 2 ∙ 𝑥 ∙ 𝑦 2
√36 ∙ √𝑥 2 ∙ √𝑦 2 ∙ √2𝑥
6𝑥𝑦√2𝑥
Description of the procedure
To simplify radical expressions you just need to evaluate the
root of the given expressions. In this case “72” is not a perfect
square, so you need to look among the factors of it for
perfect squares. For the variables you divide the exponents
by the index of the root, but remember that you do it only if
you can divide them evenly. So you will need to “split” x3 into
x2·x
After “splitting” the terms to be able to evaluate the roots,
you need to regroup the terms you are not able to evaluate,
in this case √2𝑥
Here is your final answer.
Adding and subtracting radical expressions.
Radical expression
Description of the procedure
To add or subtract radical expressions you need to use the
basic operations of algebra and combine like terms. For radical
expressions the indexes let you know which ones of them are
3
like terms. In this case the terms with index “3” ( √ ) are like
terms, and the terms with index “2” (√ ) are like terms, so
you combine them.
Regrouping to see the like terms next to each other you would
have the expression on the left.
Here is your final answer.
Rationalizing denominators in radical expressions.
Radical expression
√12𝑦 − (
𝑦
∙
√3𝑦
√3𝑦 √3𝑦
Description of the procedure
)
𝑦√3𝑦
)
√12𝑦 − (
3𝑦
Rationalizing denominators in radical expressions means to
remove any radicals from the denominator of the expression.
In this case you will need to remove the radical from the
𝑦
denominator of the term
. you will accomplish this by
√3𝑦
multiplying top and bottom of this term by √3𝑦
Then, you will able to simplify “y”
√4 ∙ 3𝑦 −
√3𝑦
3
√4 ∙ √3𝑦 −
In the term √12𝑦 you will be able to find among the factors of
12 a perfect square to simplify.
√3𝑦
3
√3𝑦
3
3 ∙ 2√3𝑦 √3𝑦
−
3
3
2√3𝑦 −
3
Now you multiply the first term by to have fractions with like
3
denominators and be able to subtract them.
6√3𝑦 √3𝑦
−
3
3
5√3𝑦
3
Here is your final answer.
Systems of equations.
System of equations
{
2𝑥 − 3𝑦 = −7
3𝑥 + 𝑦 = −5
{
(3)
2𝑥 − 3𝑦 = −7
9𝑥 + 3𝑦 = −15
11𝑥 = −22
11𝑥 = −22
11
11
𝑥 = −2
2𝑥 − 3𝑦 = −7
2(−2) − 3𝑦 = −7
−4 − 3𝑦 = −7
+4
+4
−3𝑦 = −3
−3
−3
𝑦=1
𝑥 = −2; 𝑦 = 1, or (−2,1)
Description of the procedure
To solve systems of equations you can use the elimination
method which consists on eliminating one of the variables to
solve the resulting equation for the remaining variable. First
you need to decide which variable to eliminate; then you will
need its coefficients with the same value, but with opposite
signs. In the case of the example I decided to eliminate “y” so,
I will need to multiply the second equation by 3.
Now you can eliminate the variable “y” when you add the two
equations. The resulting equation contains only one variable
now.
Now you need to solve the equation for the remaining variable
(“x”). Dividing by 11 you will get the value of “x”
Now you need to plug-in this value of “x” into any of the
original equations, and solve for the second variable (“y”). I
decided to use the first equation.
Here is your final answer.