By first multiplying the equations
to make the coefficients of either 𝑥 or 𝑦 equal, use the elimination method to
solve the given simultaneous equations to find 𝑥 and 𝑦: two 𝑥 plus three 𝑦
equals four; three 𝑥 plus four 𝑦 equals eight.
The elimination method looks
something like this. It’s when your variable terms can
cancel each other out if you add them together. So in this example that I have
drawn, two 𝑥 minus two 𝑥 would be zero and the 𝑥 term would drop out. Now back to our problem, we
actually don’t have any terms that are equal. Two 𝑥 and three 𝑦 can’t be added
together; three 𝑦 and four 𝑥 cannot be added together. What we want to do is multiply both
of these equations by numbers so that the coefficients of one of the variables,
either 𝑥 or 𝑦, are equal in both of these equations.
For example, if I multiplied the
equation on the left by three, then I would multiply three times two 𝑥, which would
give me six 𝑥. I can then multiply the equation on
the right by two; in this case we’d be multiplying two times three 𝑥, which would
give us six 𝑥.
And now, we will have two equal
variables that we could work with. So let’s go with three on the left
side. Three times two 𝑥 equals six,
three times three 𝑦 equals nine 𝑦, three times four equals 12.
And our equation on the right,
we’ll multiply by two. Two times three 𝑥 equals six 𝑥,
two times four 𝑦 equals eight 𝑦, two times eight equals 16.
Now we’ll take our first equation
and subtract our second equation from the first. Here, we’ll take six 𝑥 from six
𝑥, which would equal zero. After that, we need to subtract
eight 𝑦 from nine 𝑦. So we say nine 𝑦 minus eight 𝑦
equals one 𝑦. And finally, 12 minus 16 equals
negative four. This tells us that our 𝑦-value is
But what we want to do now is to
use one of the equations and plug in negative four for 𝑦. And our equation would look like
this: two 𝑥 plus three times negative four equals four. We multiply three times negative
four to equal negative 12. Bring down our two 𝑥 and our
four. To isolate 𝑥, we’ll need to add 12
to both sides of our equation. On the left side, we’re left with
two 𝑥. On the right side, four plus 12
Our final step to solve for 𝑥
would be to divide both sides of the equation by two. Two 𝑥 divided by two equals 𝑥; 16
divided by two equals eight. This tells us that our 𝑥-value
equals eight and our 𝑦-value equals negative four.