Question

Find the matrix B such that AB=t and A=(3 2)
(-1 3)
Hence find the point of intersection of the lines 3x+2y=10 and 3y-4x=x

Answer 1

To find the matrix B, we will solve the equation AB = t.

Given matrix A:
A = [[3, 2],
[-1, 3]]

And target matrix t:
t = [x, y]

Let matrix B be:
B = [[a, b],
[c, d]]

Then, using matrix multiplication:

AB = [[3, 2], [-1, 3]] * [[a, b], [c, d]]
= [[3a + 2c, 3b + 2d], [-a + 3c, -b + 3d]]

Setting this equal to t:
[[3a + 2c, 3b + 2d], [-a + 3c, -b + 3d]] = [x, y]

From the equation above, we can equate the corresponding elements:

Equation 1: 3a + 2c = x
Equation 2: 3b + 2d = y
Equation 3: -a + 3c = 0
Equation 4: -b + 3d = 0

Simplifying equations 3 and 4:
a = 3c
b = 3d

Substituting these values into equations 1 and 2:
3(3c) + 2c = x
3(3d) + 2d = y

Simplifying further:
9c + 2c = x
9d + 2d = y
11c = x
11d = y

Therefore, the matrix B is:
B = [[11c, 11d],
[c, d]]

To find the point of intersection of the lines 3x + 2y = 10 and 3y - 4x = x, we need to solve these two equations simultaneously.

First, let's rearrange the second equation:
3y - 4x = x
3y = 5x
y = (5/3)x

Substituting this expression for y in the first equation:
3x + 2(5/3)x = 10
3x + (10/3)x = 10
(9/3)x + (10/3)x = 10
(19/3)x = 10
19x = 30
x = 30/19

Substituting this value of x into the expression for y:
y = (5/3) * (30/19)
y = 50/19

Therefore, the point of intersection of the lines 3x + 2y = 10 and 3y - 4x = x is (30/19, 50/19).