Find the scalars m and n such that mp plus nq equals 14i plus 13j.p is 3i equals 3j and q is 4i plus 7j
what does p is 3i equals 3j mean?
I'll go with
p = 3i+3j
q = 4i+7j
So, you want to find m and n such that
m(3i+3j) + n(4i+7j) = 14i+13j
that is,
3m+4n = 14
3m+7n = 13
Now just solve for m and n.
If I got p wrong, fix it and then follow the same steps.
To find the scalars m and n, we will equate the coefficients of i and j on both sides of the equation.
Given:
mp + nq = 14i + 13j ...(1)
p = 3i + 3j ...(2)
q = 4i + 7j ...(3)
Substituting the values of p and q in equation (1), we have:
m(3i + 3j) + n(4i + 7j) = 14i + 13j
Expanding the equation, we get:
(3m + 4n)i + (3m + 7n)j = 14i + 13j
Now, we equate the coefficients of i and j separately:
For i:
3m + 4n = 14 ...(4)
For j:
3m + 7n = 13 ...(5)
Now, we have a system of two linear equations with two variables. We can solve this system to find the values of m and n.
Multiplying equation (4) by 3 and equation (5) by 4 to eliminate m, we get:
9m + 12n = 42 ...(6)
12m + 28n = 52 ...(7)
Subtracting equation (6) from equation (7), we eliminate m:
(12m + 28n) - (9m + 12n) = 52 - 42
3m + 16n = 10 ...(8)
Now, we have a new equation (8) with only n as a variable. Solving this equation will give us the value of n.
Multiplying equation (8) by 3 to eliminate m, we get:
9m + 48n = 30 ...(9)
Subtracting equation (6) from equation (9), we eliminate m:
(9m + 48n) - (9m + 12n) = 30 - 42
36n = -12
n = -12/36
n = -1/3
Substituting the value of n = -1/3 in equation (4), we can solve for m:
3m + 4(-1/3) = 14
3m - 4/3 = 14
3m = 14 + 4/3
3m = 42/3 + 4/3
3m = 46/3
m = (46/3) / 3
m = 46/9
Therefore, the scalars m = 46/9 and n = -1/3 satisfy the equation mp + nq = 14i + 13j.