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8090 [49]
4 years ago
15

Find $A$ and $B$ such that

Mathematics
2 answers:
podryga [215]4 years ago
7 0

Assume

\dfrac{3x+5}{x^2-x-42}=\dfrac{3x+5}{(x-7)(x+6)}=\dfrac A{x-7}+\dfrac B{x+6}

Combining the fractions on the right hand side gives

\dfrac{3x+5}{(x-7)(x-6)}=\dfrac{A(x-6)+B(x-7)}{(x-7)(x-6)}

The fractions will be equal as long as the their numerator are equal:

3x+5=A(x-6)+B(x-7)

3x+5=(A+B)x+(-6A-7B)

Polynomials are equal to one another if the coefficients of like terms are equal:

\begin{cases}A+B=3\\-6A-7B=5\end{cases}

Solving this, you'd get A=26 and B=-23, so that your answer would be (26, -23).

- - -

Instead of solving the system of equations above, there is a trick that involves picking x so that some terms disappear and solving for either A,B is much faster.

At the point where we have

3x+5=A(x-6)+B(x-7)

notice that setting x=7 will eliminate B. Doing so, we get

3(7)+5=A(7-6)\implies26=A

while setting x=6 would give

3(6)+5=B(6-7)\implies23=-B\implies B=-23

tester [92]4 years ago
6 0

Answer:

We factor the denominator in the left-hand side to get

\[\frac{3x+5}{(x-7)(x+6)}= \frac{A}{x - 7} + \frac{B}{x + 6}.\]

We then multiply both sides by

$(x - 7)(x + 6)$, to get \[3x + 5 = A(x + 6) + B(x - 7).\]

We can solve for $A$ and $B$ by substituting suitable values of $x$.

For example, setting $x = 7$, the equation becomes $26 = 13A$,

so $A = 2$.

Setting $x = -6$, the equation becomes $-13 = -13B$,

so $B = 1$

Therefore, $(A,B) = \boxed{(2,1)}$.

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True or false? If false give counterexample The product of a rational number and an integer is not an integer ​
Rzqust [24]

Answer:

False

Step-by-step explanation:

Required

State if the product of rational numbers and integer is an integer

The statement is false and the proof is as follows

Literally, rational numbers are decimal numbers that can be represented as a fraction of two integers;

Take for instance: 0.2, 0.5, 2.25, etc.

When any of these numbers is multiplied by an integer, the resulting number can take any of two forms;

1. It can result to an integer:

<em>For instance;</em>

0.2 * 5 = 1

0.5 * 4 = 2

2.25 * 8 = 18

2. It can result in a decimal number

<em>For instance;</em>

0.2 * 3 = 0.6

0.5 * 5 = 2.5

2.25 * 7 = 15.75

From (1) above, we understand that the product can result in an integer.

Hence, the statement is false

4 0
4 years ago
Please show all work.
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See the attached picture:

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3 years ago
Which symbol makes the sentence true?
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5 0
4 years ago
How many numbers are divisible by 10 from 5 to 200?
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5 0
4 years ago
A video store charges the same fee each day a video is returned late. Craig returned a video 4 days late and paid a $3 fee. Writ
vova2212 [387]

Answer:

$5.25 if returned 7 days late

Step-by-step explanation:

So we know that 4 days late is 3 dollars

We can find the unit rate

3/4 is 0.75

So 75 cents for every late day

Now we just multiply that by 7

0.75*7=5.25

so 7 days, 5.25

**porportional relationship equation**

\frac{3dollars}{4 days}=\frac{X}{7 days}

Now cross multiply

21=4x

x=5.25

Same results

4 0
3 years ago
Read 2 more answers
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