Use the elimination method to solve the given simultaneous equations. Four 𝑥 minus two 𝑦 equals four and five 𝑥 plus three 𝑦 equals 16.
Our first step is to either make the coefficients of 𝑦 or the coefficients of 𝑥 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 𝑥 the same. And in this question, they would both be 20𝑥.
In this question, we are going to use the first method. Multiplying the top equation by three gives us 12𝑥 minus six 𝑦 equals 12. And multiplying the bottom equation by two gives us 10𝑥 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 𝑥 terms gives us 22𝑥, and adding the numbers on the right-hand side gives us 44. Dividing both sides of this equation by 22 gives us an 𝑥-value equal to two, 𝑥 equals two.
We now need to substitute this value of 𝑥, 𝑥 equals two, into one of the equations. We can choose any of the four equations here: four 𝑥 minus two 𝑦 equals four, five 𝑥 plus three 𝑦 equals 16, or the two below that. In this case, we’re gonna choose five 𝑥 plus three 𝑦 equals 16, as all our terms are positive. Substituting in 𝑥 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 𝑥 minus two 𝑦 equals four and five 𝑥 plus three 𝑦 equals 16 are: 𝑥 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 𝑥-coordinate would be two and the 𝑦-coordinate would also be two.