Video: Solving Simultaneous Using Elimination Where One of the Equations Needs to be Multiplied

Using elimination, solve the simultaneous equations 3π‘₯ + 7𝑦 = 34, 9π‘₯ + 10𝑦 = 91.

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Video Transcript

Using elimination, solve the simultaneous equations three π‘₯ plus seven 𝑦 equals 34 and nine π‘₯ plus 10 𝑦 equals 91.

Our first step is to make either the π‘₯ or the 𝑦 coefficients the same. In this case, the easiest way to do this is to multiply the first equation by three. Multiplying three π‘₯ by three gives us nine π‘₯. Multiplying seven 𝑦 by three gives us 21 𝑦. And 34 multiplied by three is 102.

If we then subtract equation two from equation one, the π‘₯ terms cancel as nine π‘₯ minus nine π‘₯ is zero. 21 𝑦 minus 10 𝑦 is equal to 11 𝑦. And 102 minus 91 is equal to 11. Dividing both sides of this equation by 11 gives us an answer for 𝑦 equal to one.

In order to work out our value for π‘₯, we need to substitute this value for 𝑦 into one of the equations. In this case, I’m going to substitute 𝑦 equals one into equation two. Substituting in this value for 𝑦 gives us nine π‘₯ plus 10 multiplied by one equals 91.

As 10 multiplied by one is 10, we’re left with nine π‘₯ plus 10 equals 91. We can then subtract 10 from both sides of the equation, leaving us nine π‘₯ is equal to 81. And finally dividing both sides of this equation by nine leaves as a value for π‘₯ equal to nine.

Therefore, the solution to the simultaneous equations three π‘₯ plus seven 𝑦 equals 34 and nine π‘₯ plus 10 𝑦 equals 91 are 𝑦 equals one and π‘₯ equals nine.

We could check these two answers by substituting the values of π‘₯ and 𝑦 back into the other equation, number one. Nine multiplied by nine plus 21 multiplied by one is equal to 102. Therefore, our solution is correct.

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