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Given two matrices, solve for the unknown variables in each.

Mathematics

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2 parts
a

x=1,y=2,z=3x=1, y=2, z=3

b

x=−3,y=8,p=5,q=−2x=-3, y=8, p=5, q=-2

Step-by-step solution

8 steps

Part a· 4 steps

To find the values of x, y, and z, we equate the corresponding elements of the given matrices

(x+yz−xy+2zx)=(3281)\begin{pmatrix} x+y & z-x \\ y+2z & x \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 8 & 1 \end{pmatrix}

This gives us the following system of equations:

  1. x+y=3x+y = 3
  2. z−x=2z-x = 2
  3. y+2z=8y+2z = 8
  4. x=1x = 1

  1. From equation (4), we directly get the value of x.

    x=1x = 1

  2. Substitute x=1x=1 into equation (1).

    1+y=31+y = 3 y=3−1y = 3-1 y=2y = 2

  3. Substitute x=1x=1 into equation (2).

    z−1=2z-1 = 2 z=2+1z = 2+1 z=3z = 3

  4. Verify these values with equation (3).

    y+2z=2+2(3)=2+6=8y+2z = 2+2(3) = 2+6 = 8

    The values satisfy all equations.

    The values are: x=1,y=2,z=3\boxed{x=1, y=2, z=3}

Part b

x=−3,y=8,p=5,q=−2x=-3, y=8, p=5, q=-2

To find the values of x, y, p, and q, we equate the corresponding elements of the given matrices

(p+15−3q)=(6y−3x−2)\begin{pmatrix} p+1 & 5 \\ -3 & q \end{pmatrix} = \begin{pmatrix} 6 & y-3 \\ x & -2 \end{pmatrix}

This gives us the following system of equations:

  1. p+1=6p+1 = 6
  2. 5=y−35 = y-3
  3. −3=x-3 = x
  4. q=−2q = -2

  1. Solve for p from equation (1).

    p+1=6p+1 = 6 p=6−1p = 6-1 p=5p = 5

  2. Solve for y from equation (2).

    5=y−35 = y-3 y=5+3y = 5+3 y=8y = 8

  3. From equation (3), we directly get the value of x.

    x=−3x = -3

  4. From equation (4), we directly get the value of q.

    q=−2q = -2

    The values are: x=−3,y=8,p=5,q=−2\boxed{x=-3, y=8, p=5, q=-2}

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