Use the elimination method to solve the given simultaneous equations: five 𝑥 plus 𝑦 equals 20 and four 𝑥 plus five 𝑦 equals 37.
Our first step is to make the coefficients of 𝑥 or the coefficients of 𝑦 the same. In this case, multiplying the top equation by five makes both of the 𝑦-coefficients five. Multiplying all the terms in this top equation leaves us with 25𝑥 plus five 𝑦 equals 100.
At this stage, we’re going to leave the second equation as it is, as both of the 𝑦-coefficients are five. Subtracting these two equations eliminates the 𝑦s, as five 𝑦 minus five 𝑦 equals zero. In the same way, when we subtract the 𝑥-values, we end up with 21𝑥. And on the right-hand side, 100 minus 37 is 63. Dividing both sides of this equation by 21 gives us an 𝑥-value equal to three.
In any pair of simultaneous equations, we need to work out the 𝑥-value and also the 𝑦-value. To do this, we’ll substitute 𝑥 equals three into one of the equations. In this case, we’re going to substitute it into the equation five 𝑥 plus 𝑦 equals 20, but we could quite easily pick any one of the other equations. Five multiplied by three is 15. Therefore, 15 plus 𝑦 equals 20. Subtracting 15 from both sides of this equation gives us a final 𝑦-value equal to five.
Therefore, the solution to the pair of simultaneous equations, five 𝑥 plus 𝑦 equals 20 and four 𝑥 plus five 𝑦 equals 37, our 𝑥 equals three and 𝑦 equals five. We can check these answers by substituting the values into the other equation, four 𝑥 plus five 𝑦 equals 37. Four multiplied by three is 12; five multiplied by five is 25; 12 plus 25 equals 37.
As the values 𝑥 equals three and 𝑦 equals five satisfy both equations, we know that our answers must be correct. This question could also be solved graphically on a coordinate axis. The point of intersection of the two lines would have coordinate or ordered pair three, five. The 𝑥-coordinate would be three and the 𝑦-coordinate would be five.