Video: Solving Simultaneous Equations by Elimination

Use the elimination method to solve the given simultaneous equations 4π‘₯ βˆ’ 2𝑦 = 4, 5π‘₯ + 3𝑦 = 16.

03:10

Video Transcript

Use the elimination method to solve the given simultaneous equations. Four π‘₯ minus two 𝑦 equals four and five π‘₯ plus three 𝑦 equals 16.

Our first step is to either make the coefficients of 𝑦 or the coefficients of π‘₯ the same. We could do this by multiplying the top equation by three and the bottom equation by two. This would make the coefficient of the 𝑦 terms six. Alternatively, we could multiply the top equation by five and the bottom equation by four. This would make the coefficient of π‘₯ the same. And in this question, they would both be 20π‘₯.

In this question, we are going to use the first method. Multiplying the top equation by three gives us 12π‘₯ minus six 𝑦 equals 12. And multiplying the bottom equation by two gives us 10π‘₯ plus six 𝑦 equals 32. We now need to eliminate the 𝑦 terms. We are going to do this by adding our two equations. Negative six 𝑦 plus positive six 𝑦 gives us zero. Adding the π‘₯ terms gives us 22π‘₯, and adding the numbers on the right-hand side gives us 44. Dividing both sides of this equation by 22 gives us an π‘₯-value equal to two, π‘₯ equals two.

We now need to substitute this value of π‘₯, π‘₯ equals two, into one of the equations. We can choose any of the four equations here: four π‘₯ minus two 𝑦 equals four, five π‘₯ plus three 𝑦 equals 16, or the two below that. In this case, we’re gonna choose five π‘₯ plus three 𝑦 equals 16, as all our terms are positive. Substituting in π‘₯ equals two gives us five multiplied by two plus three 𝑦 equals 16. As five multiplied by two is 10, this can be rewritten as 10 plus three 𝑦 equals 16. We then need to balance this equation to work out our value of 𝑦. Firstly, subtracting 10 from both sides of the equation leaves us with three 𝑦 equals six as 16 minus 10 equals six. Finally, dividing both sides of this equation by three gives us a 𝑦-value also equal to two.

Therefore, the solution to the pair of simultaneous equations four π‘₯ minus two 𝑦 equals four and five π‘₯ plus three 𝑦 equals 16 are: π‘₯ equals two and 𝑦 equals two.

This could also be demonstrated on a coordinate axes by plotting the two linear equations. The point of intersection would be the ordered pair or coordinate two, two as the π‘₯-coordinate would be two and the 𝑦-coordinate would also be two.

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