Rational Equations Using Proportions
Key Questions
-
Answer:
Solving proportions is like solving fractions:
a:b -> c:d can be rewritten asa/b = c/d and now you can solve for any of the variables.See an example below:
Explanation:
3 to 4 is like what to 16?
This can be rewritten as a proportion:
3:4 -> c:16 Which can be rewritten as:
3/4 = c/16 Which can be solved as:
color(red)(16) xx 3/4 = color(red)(16) xx c/16 cancel(color(red)(16))color(red)(4) xx 3/color(red)(cancel(color(black)(4))) = cancel(color(red)(16)) xx c/color(red)(cancel(color(black)(16))) 12 = c c = 12 3 to 4 is like 12 to 16?
-
We multiply the numerator of each (or one) side by the denominator of the other side.
For example, if I have want to solve for
x for the following equation:x/5=3/4 I can use cross-multiplication, and the equation becomes:
x*4=3*5 4x=15 x=15/4=3.75 -
A proportion is a statement that two ratios are equal to each other.
For example3/6=5/10 (We sometimes read this "3 is to 6 as 5 is to 10".)There are
4 'numbers' (really number places) involved. If one or more of those 'numbers' is a polynomial, then the proportion becomes a rational equation.For example:
(x-2)/2=7/(x+3) ("x-2 is to 2 as 7 is to x+3").Typically, once they show up, we want to solve them. (Find the values of
x that make them true.)In the example we would "cross multiply" or multiply both sides by the common denominator (either description applies) to get:
(x-2)(x+3)=2*7 . Which is true exactly when
x^2+x-6=14 Which in turn, is equivalent to
x^2+x-20=0 (Subtract 14 on both sides of the equation.)
Solve by factoring(x+5)(x-4)=0
so we needx+5=0 orx-4=0 the first requires
x=-5 and the secondx=4 .Notice that we can check our answer:
(-5-2)/2=-7/2 and7/(-5+3)=7/-2=-7/2 . So the ratios on both sides are equal and the statement is true.
Questions
Rational Equations and Functions
-
Inverse Variation Models
-
Graphs of Rational Functions
-
Division of Polynomials
-
Excluded Values for Rational Expressions
-
Multiplication of Rational Expressions
-
Division of Rational Expressions
-
Addition and Subtraction of Rational Expressions
-
Rational Equations Using Proportions
-
Clearing Denominators in Rational Equations
-
Surveys and Samples