Solve the simultaneous equations
four 𝑥 plus three 𝑦 is equal to 10; negative three 𝑥 plus five 𝑦 is equal to
We’ll begin just by labelling the
equations as one and two so that we can refer to them more easily. We’re going to use the elimination
method to answer this question, which means we want to first eliminate either the 𝑥
or the 𝑦 terms. We’ll eliminate the 𝑥 terms, which
means we need to make the coefficients of 𝑥 the same in both equations.
As the coefficients are currently
four and negative three, we need to find the lowest common multiple of four and
three, which is 12. To get from four to 12, we have to
multiply by three. So we multiply the whole of
equation one by three, giving 12𝑥 plus nine 𝑦 is equal to 30. To get from three to 12, we have to
multiply by four. So I multiply the whole of equation
two by four, giving negative 12𝑥 plus 20𝑦 is equal to negative 88. Now, we notice that the
coefficients of 𝑥 are almost the same, but one is positive and one is negative.
An equally valid approach would
have been to eliminate the 𝑦 terms. So we could have multiplied
equation one by five and equation two by three. Both equations will then have
15𝑦. Assuming we perform all of the next
stages correctly, this method would lead us to the same answer.
Returning to our equations, we know
that because the 𝑥 terms have different signs, we eliminate the 𝑥s by adding the
two equations together. 12𝑥 plus negative 12𝑥 gives zero
𝑥, nine 𝑦 plus 20𝑦 gives 29𝑦, and 30 plus negative 88 gives negative 58. So we have 29𝑦 is equal to
negative 58 and we’ve eliminated the 𝑥 terms. To find the value of 𝑦, we need to
divide both sides of the equation by 29, giving 𝑦 is equal to negative two.
So we’ve solved the equations for
𝑦. And now, we need to find the value
of 𝑥. To do so, we need to substitute the
value of 𝑦 that we’ve just found into any of the four equations. It doesn’t matter which one we
choose. Let’s substitute into equation one,
which was four 𝑥 plus three 𝑦 is equal to 10. This gives four 𝑥 plus three
multiplied by negative two is equal to 10.
Three multiplied by negative two is
negative six. So we have four 𝑥 minus six is
equal to 10. And we can solve this equation to
find the value of 𝑥. Adding six to both sides gives four
𝑥 is equal to 16. And dividing both sides by four
gives 𝑥 is equal to four. The solution to the simultaneous
equations is 𝑥 equals four, 𝑦 equals negative two.
A sensible step to do at the end of
this question would be to check our answer by substituting the values we found for
𝑥 and 𝑦 into the second equation. Negative three 𝑥 would become
negative three multiplied by four and five 𝑦 would become five multiplied by
negative two. This gives negative 12 minus 10
which simplifies to negative 22. This is what we’re expecting
negative three 𝑥 plus five 𝑦 to be equal to. And therefore, we can be confident
in our answer.