Question Video: Solving Simultaneous Equations by Elimination

Use the elimination method to solve the given simultaneous equations 5π‘₯ + 𝑦 = 20, 4π‘₯ + 5𝑦 = 37.

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Video Transcript

Use the elimination method to solve the given simultaneous equations: five π‘₯ plus 𝑦 equals 20 and four π‘₯ plus five 𝑦 equals 37.

Our first step is to make the coefficients of π‘₯ 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 25π‘₯ 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 π‘₯-values, we end up with 21π‘₯. And on the right-hand side, 100 minus 37 is 63. Dividing both sides of this equation by 21 gives us an π‘₯-value equal to three.

In any pair of simultaneous equations, we need to work out the π‘₯-value and also the 𝑦-value. To do this, we’ll substitute π‘₯ equals three into one of the equations. In this case, we’re going to substitute it into the equation five π‘₯ 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 π‘₯ plus 𝑦 equals 20 and four π‘₯ plus five 𝑦 equals 37, our π‘₯ equals three and 𝑦 equals five. We can check these answers by substituting the values into the other equation, four π‘₯ 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 π‘₯ 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 π‘₯-coordinate would be three and the 𝑦-coordinate would be five.

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