Video Transcript
Using elimination, solve the
simultaneous equations four 𝑥 plus six 𝑦 equals 40 and three 𝑥 plus seven 𝑦
equals 40.
In the two equations we’ve been
given, the coefficients of 𝑥 are different and the coefficients of 𝑦 are also
different, which means we can’t eliminate one variable by just adding or
subtracting the equations together. We also notice that neither of
the coefficients of 𝑥 are factors of the other and neither of the coefficients
of 𝑦 are factors of each other. We want to create equations in
which the coefficients of either variable are the same or at least the same but
with different signs. So how are we going to do
this?
Well, we’re going to have to
multiply both equations by some constant. I’m going to choose to multiply
equation 1 by three and equation 2 by four because this will create 12𝑥 in each
equation. We could also have chosen to
multiply equation 1 by seven and equation 2 by six as this would create 42𝑦 in
each equation. It doesn’t matter which
variable we choose to eliminate.
Now that we have our two new
equations, we have 12𝑥 in each. And as the signs are the same,
we can eliminate the 𝑥-variable by subtracting. I’m actually going to subtract
the top equation from the bottom one because the coefficient of 𝑦 is greater in
the second equation. We have 12𝑥 minus 12𝑥, which
cancels out; 28𝑦 minus 18𝑦, which gives 10𝑦; and 160 minus 120, which is
40. So we’ve eliminated the
𝑥-variable from our equation. We can then solve for 𝑦 by
dividing both sides of this equation by 10, giving 𝑦 equals four.
To solve for 𝑥, we substitute
this value of 𝑦 into any of our four equations. I’m going to choose equation
1. Doing so gives a
straightforward linear equation for 𝑥 which we can solve by subtracting 24 and
dividing by four to give 𝑥 equals four. So we have our solution. Both 𝑥 and 𝑦 are equal to
four. As always, we should check our
solution by substituting our values into any of the other equations. I’ve used equation 2. And it confirms that our
solution is correct.
The key step in this question
then was to multiply both equations by some number to create the same
coefficient of one of the variables. We could then use the method of
elimination to eliminate this variable and solve our simultaneous equations.
