College Algebra Unit 2 on 03 March 2025 at 12.05.42 PM

Solving for Variables

• Start by solving for the variable k.

  • Example: If 2k = 32, divide both sides by 2.

  • Result: k = 16 (k was miscalculated in the conversation).

Directly and Inversely Proportional Relationships

• Directly Proportional: When one variable increases, the other increases proportionally. Example: If y = kx.

  • If x = 3, find y to be 6 (as mentioned in the video).

• Inversely Proportional: Described by the equation y = k/x or y = 1/(kx).

Multiple Variables in Equations

• Consider equations with joint variation. For instance,

  • Example: y varies jointly with x and z, and inversely with w².

  • Example equation: y = k * (x * z) / (w²)

Understanding Systems of Equations

• When row reducing a matrix,

  • If the last row indicates 0 = 0, the system has infinitely many solutions.

Finding Solutions

• Solving for x, y, and z in terms of each other:

  • If given equations, convert and simplify.

  • Example:

    • From the row x + 0y - z = -2 solving gives x = z - 2.

    • Simplifying y leads to y = -2z + 3.

General Solution Set

• A solution set in terms of z is presented as:

  • (z - 2, -2z + 3, z)

  • Where any z value chosen alters the results.

Example Problem: Stamp Purchases

• Define your variables:

  • x = number of 49¢ stamps

  • y = number of 21¢ stamps

• Create system of equations:

  • x + y = 200 (total number of stamps)

  • 0.49x + 0.21y = 86.80 (total value of the stamps)

• Write down the augmented matrix to solve:

  • Matric format:

    [\begin{bmatrix} 1 & 1 & | & 200 \ 0.49 & 0.21 & | & 86.80\end{bmatrix}]

• Row reduce the matrix

  • Look for ones on the diagonal ➔ indicates unique solutions.

Solution Interpretation

• Example solution: x = 160 (49¢ stamps), y = 40 (21¢ stamps).

Miscellaneous Notes

• Reference for systems of equations—particularly 2x2 and 3x3 scenarios is vital.

• Always check calculations for clarity and ensure that the dimensions match in matrix setup.