9.3 Simultaneous equations concepts -- elimination

Solving simultaneous equations by elimination

$\mathrm{3x}+\mathrm{4y}=1$

$\mathrm{2x}-y=8$

• 1

  The first question is: how do we get one of these variables to have the same coefficient in both equations ? To get the $x$ coefficients to be the same, we would have to multiply the top equation by 2 and the bottom by 3. It is much easier with $y$ ; if we simply multiply the bottom equation by 4, then the two $y$ values will both be multiplied by 4.
  • $\mathrm{3x}+\mathrm{4y}=1$
  • $\mathrm{8x}-\mathrm{4y}=\text{32}$

• 2

  Now we either add or subtract the two equations. In this case, we have $\mathrm{4y}$ on top, and $-\mathrm{4y}$ on the bottom; so if we add them, they will cancel out. (If the bottom had $a$ $+\mathrm{4y}$ we would have to subtract the two equations to get the " $y$ "s to cancel.)
  • $\text{11}x+\mathrm{0y}=\text{33}$

• 3-4

  Once again, we are left with only one variable. We can solve this equation to find that $x=3$ and then plug back in to either of the original equations to find $y=-2$ as before.