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Find the values of a and b that turn equation into identity

Find the values of a and b that turn equation into identity-example-1
User Degr
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2 Answers

4 votes
(2x+5) / (x-8)(2x+1) = (a/x-8) + (b/2x+1)

Let's find the values of a and b by equating the numerators:

2x + 5 = a(2x+1) + b(x-8)

2x + 5 = 2ax + a + bx - 8b

now we will get X coefficient

2 = 2a + b

and now we will get the constant term

5 = a - 8b

so

eq1:.

2a + b = 2

and
eq2:.

a - 8b = 5

from the first equation

2 = 2a + b

2 - b = 2a

-b = 2a - 2

b = -2a + 2

and now we will substitute equation 2

a - 8b = 5

a - 8 ( -2a + 2) = 5

a + 16a - 16 = 5

a + 16a = 5 + 16

17a = 21

a = ²¹/₁₇ = 1.24

1.24 - 8b = 5

-8b = 5 - 1.24

-8b = 3.76

b = -0.47
User Unlimitednzt
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8.3k points
2 votes

Answer:


\sf a = (21)/(17)


\sf b = -(8)/(17)

Explanation:

To find the values of
\sf a and
\sf b such that the given expression becomes an identity, we can use partial fraction decomposition. The expression in question is:


\sf (2x + 5)/((x - 8)(2x + 1))

The denominator can be factored as:


\sf (x - 8)(2x + 1).

Now, we express the given fraction as the sum of two fractions with unknown values
\sf a and
\sf b:


\sf (2x + 5)/((x - 8)(2x + 1)) = (a)/(x - 8) + (b)/(2x + 1)

To find
\sf a and
\sf b, we can clear the denominators by multiplying both sides of the equation by the common denominator
\sf (x - 8)(2x + 1). This gives:


\sf 2x + 5 = a(2x + 1) + b(x - 8)

Now, we can solve for
\sf a and
\sf b by comparing coefficients. Let's expand the right side:


\sf 2x + 5 = 2ax + a + bx - 8b

Now, group like terms:


\sf 2x + 5 = (2a + b)x + (a - 8b)

By comparing coefficients, we can write a system of equations:


\sf 2a + b = 2


\sf a - 8b = 5

Solving this system will give us the values of
\sf a and
\sf b that turn the equation into an identity. Let's solve the system:


\sf 2a + b = 2


\sf a - 8b = 5

Multiplying the second equation by 2 to match coefficients:


\sf 2a + b = 2


\sf 2a - 16b = 10

Subtracting the first equation from the second:


\sf 2a - 16b -2a-b = 10-2


\sf -17b = 8


\sf b = -(8)/(17)

Now, substitute the value of
\sf b back into one of the original equations. Using the first equation:


\sf 2a - (8)/(17) = 2

Multiply both sides by 17 to get rid of the fraction:


\sf 2a\cdot 17 - (8)/(17) \cdot 17= 2\cdot 17


\sf 34a - 8 = 34


\sf 34a = 42


\sf a = (21)/(17)

So, the values of
\sf a and
\sf b that turn the given expression into an identity are:


\sf a = (21)/(17)


\sf b = -(8)/(17)

User Sharda Singh
by
7.6k points

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