Module 8: Linear Equations and Linear Systems

Example:

${"version":"1.1","math":"3x+4y=70"}$ and ${"version":"1.1","math":"3x+2y=50"}$

1. Subtract both equations to eliminate a variable by vertical placement. Keep track of the signs in front of each term and account for them in the subtraction. The goal is to eliminate a variable by getting a difference of 0. Pick a variable you wish to eliminate and find the lowest common multiple (LCM) between the coefficients bearing that variable. Equate the coefficients of these terms with LCM and adjust the rest of the equation accordingly. Subtract to eliminate a variable, then isolate for the remaining variable.

   ${"version":"1.1","math":"3x+4y=70"}$

   ${"version":"1.1","math":"-3x+2y=50"}$

   ${"version":"1.1","math":"-0x+2y=20"}$

   ${"version":"1.1","math":"y=10"}$

   This example is convenient as the coefficients of x are both 3. By subtracting 3x and 3 x, we get 0. If you wanted to eliminate y though, notice that the coefficient for y are not the same. The equations would need to be adjusted so that the coefficients of y are the same. In this case, the LCM is 4. The top equation doesn't need to be adjusted, but the bottom equation needs to be adjusted by multiplying every term by 2.

2. Using the newfound variable, pick a linear equation of the two and plug it in to solve for the second variable.

   ${"version":"1.1","math":"3x+2(10)=50"}$

   ${"version":"1.1","math":"3x=30"}$

   ${"version":"1.1","math":"x=10\rightarrow POI:(10,10)"}$