The binary operation *on the set R of all real numbers is defined as a*b=2a+3b-5.
a)find the inverse element of*
b)show whether or not *is commutative
c)find -3 *1/4
Is * commutative?
a*b = 2a+3b-5
b*a = 2b+3a-5
I think you can see that in general, * is not commutative.
In order to have an inverse, you need an identity c, such that
a*c = a
For example,
the additive identity is 0, so the additive inverse of a is -a
the multiplicative identity is 1, so the mult. inverse of a is 1/a
so, we need
2a+3c-5 = a
c = (5-a)/3
So, use that to find the inverse of a.
a) Sure, let's find the inverse element of *. For every real number a, the inverse element of a under * can be found by solving the equation a * x = x * a = 0, where 0 represents the identity element. Let's solve it together:
a * x = 2a + 3x - 5
x * a = 2x + 3a - 5
To find the inverse, let's set both equations equal to zero:
2a + 3x - 5 = 0
2x + 3a - 5 = 0
Now, let's solve this system of linear equations:
Let's multiply the first equation by 2 and the second equation by 3 to eliminate the "a" coefficient:
4a + 6x - 10 = 0
6a + 9x - 15 = 0
Let's simplify both equations:
4a + 6x = 10
6a + 9x = 15
Now, let's eliminate "x" by multiplying the first equation by -3 and the second equation by 2:
-12a - 18x = -30
12a + 18x = 30
Now, let's add the equations together:
-12a - 18x + 12a + 18x = -30 + 30
0 = 0
As we can see, the sum of both equations results in 0. Therefore, any real number can be an inverse element under *.
b) Now, let's determine whether * is commutative or not. For * to be commutative, the following condition must hold: a * b = b * a for any real numbers a and b.
Let's check this condition:
Assume a = 1 and b = 2:
1 * 2 = 2(1) + 3(2) - 5 = 2 + 6 - 5 = 3
Now, assume a = 2 and b = 1:
2 * 1 = 2(2) + 3(1) - 5 = 4 + 3 - 5 = 2
Since 1 * 2 ≠ 2 * 1, we can conclude that * is not commutative.
c) Finally, let's find -3 * 1/4:
-3 * 1/4 = 2(-3) + 3(1/4) - 5
= -6 + 3/4 - 5
= -6 - 19/4
= -24/4 - 19/4
= -43/4
So, -3 * 1/4 equals -43/4.
a) To find the inverse element of *, we need to find a number x such that for any real number a, a * x = x * a = 0 (the identity element).
Let's find the inverse element x. We have:
a * x = 2a + 3x - 5
Comparing this to the identity element equation, we can see that:
2a + 3x - 5 = 0
Rearranging the terms, we get:
3x = 5 - 2a
Dividing both sides by 3, we obtain:
x = (5 - 2a) / 3
Therefore, the inverse element of * is (5 - 2a) / 3.
b) To show whether * is commutative, we need to verify if a * b = b * a for any two real numbers a and b.
Let's consider a = 2 and b = 3:
a * b = 2 * 2 + 3 * 3 - 5 = 4 + 9 - 5 = 8
b * a = 2 * 3 + 3 * 2 - 5 = 6 + 6 - 5 = 7
Since a * b is not equal to b * a for a = 2 and b = 3, we can conclude that * is not commutative.
c) To find -3 * 1/4, we substitute a = -3 and b = 1/4 into the definition of *:
-3 * 1/4 = 2 * (-3) + 3 * (1/4) - 5
Simplifying, we get:
-3 * 1/4 = -6 + 3/4 - 5
Combining like terms, we have:
-3 * 1/4 = -31/4
Therefore, -3 * 1/4 is equal to -31/4.
a) To find the inverse element of *, we need to find an element b such that a * b = b * a = e, where e is the identity element for the operation *.
Let's set a * b = e and substitute the given expression:
2a + 3b - 5 = e.
In this case, e would be the additive identity element, which is 0. So, we have:
2a + 3b - 5 = 0.
Solving this equation for b, we get:
3b = 5 - 2a,
b = (5 - 2a) / 3.
Hence, the inverse element of * is (5 - 2a) / 3.
b) To check whether * is commutative, we need to verify if a * b = b * a for any pair of real numbers a and b.
Let's substitute the values of a and b into the definition of *:
a * b = 2a + 3b - 5,
b * a = 2b + 3a - 5.
Now, let's compare the two expressions:
2a + 3b - 5 ≠ 2b + 3a - 5.
Since the two expressions are not equal, we can conclude that * is not commutative.
c) To evaluate -3 * (1/4), we substitute -3 for a and 1/4 for b in the definition of *:
-3 * (1/4) = 2(-3) + 3(1/4) - 5.
Simplifying the expression, we get:
-3 * (1/4) = -6 + 3/4 - 5.
To perform arithmetic with mixed fractions, we first convert the mixed numbers into improper fractions:
-3 * (1/4) = -6 + (3/4) - 5.
Now, we find a common denominator for the fractions:
-3 * (1/4) = -6 + (3/4) - (20/4).
Combining the fractions and whole numbers, we get:
-3 * (1/4) = -22/4.
Finally, simplifying the fraction, we have:
-3 * (1/4) = -11/2.