Question Video: Solving Simultaneous Equations by Elimination | Nagwa Question Video: Solving Simultaneous Equations by Elimination | Nagwa

# Question Video: Solving Simultaneous Equations by Elimination Mathematics • Third Year of Preparatory School

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

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### Video Transcript

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

Our first step is to make the coefficients of π₯ or the coefficients of π¦ the same. In this case, multiplying the top equation by five makes both of the π¦-coefficients five. Multiplying all the terms in this top equation leaves us with 25π₯ plus five π¦ equals 100.

At this stage, weβre going to leave the second equation as it is, as both of the π¦-coefficients are five. Subtracting these two equations eliminates the π¦s, as five π¦ minus five π¦ 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 π¦-value. To do this, weβll substitute π₯ equals three into one of the equations. In this case, weβre going to substitute it into the equation five π₯ plus π¦ equals 20, but we could quite easily pick any one of the other equations. Five multiplied by three is 15. Therefore, 15 plus π¦ equals 20. Subtracting 15 from both sides of this equation gives us a final π¦-value equal to five.

Therefore, the solution to the pair of simultaneous equations, five π₯ plus π¦ equals 20 and four π₯ plus five π¦ equals 37, our π₯ equals three and π¦ equals five. We can check these answers by substituting the values into the other equation, four π₯ plus five π¦ 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 π¦ 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 π¦-coordinate would be five.

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