A binary operation \circ defined on the set of real numbers \mathbb{R} is given by a \circ… — JAMB UTME

A binary operation $\circ$ defined on the set of real numbers $\mathbb{R}$ is given by $a \circ b = 3a + 2b - 1$ . What is the value of $x$ for which $(2 \circ x) \circ 3 = 26$ ?

$1$ Cevap
B
$\frac{11}{6}$
C
$0$
D
$\frac{5}{6}$

Cevap

The value of

x

for which

(2 \circ x) \circ 3 = 26

1

Applying the binary operation rule

a \circ b = 3a + 2b - 1

to the inner expression gives

2 \circ x = 2x + 5

. Then applying the rule to

(2x + 5) \circ 3

gives

3(2x + 5) + 2(3) - 1 = 6x + 20

. Equating

6x + 20 = 26

yields

6x = 6

, giving

x = 1

Adım Adım Çözüm

Evaluate the inner binary operation expression

2 \circ x

using the rule

a \circ b = 3a + 2b - 1

2 \circ x = 3(2) + 2(x) - 1 = 6 + 2x - 1 = 2x + 5

The definition of the binary operation replaces

a

with

2

and

b

with

x

Substitute the result

(2x + 5)

into the outer binary operation expression

(2 \circ x) \circ 3

(2x + 5) \circ 3 = 3(2x + 5) + 2(3) - 1 = 6x + 15 + 6 - 1 = 6x + 20

Apply the rule

a \circ b = 3a + 2b - 1

where

a = 2x + 5

and

b = 3

Equate the simplified expression to

26

and solve for

x

6x + 20 = 26 \implies 6x = 6 \implies x = 1

Setting the calculated expression equal to the given value allows isolation of

x

Anahtar Kavram

Evaluation of non-commutative composite binary operations

Soru

Cevap

Adım Adım Çözüm

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