Question Video: Solving Simultaneous Using Elimination, Where Both of the Equations Needs to be Multiplied Mathematics • 8th Grade

Video Transcript

Using elimination, solve the simultaneous equations four 𝑥 plus six 𝑦 equals 40 and three 𝑥 plus seven 𝑦 equals 40.

In the two equations we’ve been given, the coefficients of 𝑥 are different and the coefficients of 𝑦 are also different, which means we can’t eliminate one variable by just adding or subtracting the equations together. We also notice that neither of the coefficients of 𝑥 are factors of the other and neither of the coefficients of 𝑦 are factors of each other. We want to create equations in which the coefficients of either variable are the same or at least the same but with different signs. So how are we going to do this?

Well, we’re going to have to multiply both equations by some constant. I’m going to choose to multiply equation 1 by three and equation 2 by four because this will create 12𝑥 in each equation. We could also have chosen to multiply equation 1 by seven and equation 2 by six as this would create 42𝑦 in each equation. It doesn’t matter which variable we choose to eliminate.

Now that we have our two new equations, we have 12𝑥 in each. And as the signs are the same, we can eliminate the 𝑥-variable by subtracting. I’m actually going to subtract the top equation from the bottom one because the coefficient of 𝑦 is greater in the second equation. We have 12𝑥 minus 12𝑥, which cancels out; 28𝑦 minus 18𝑦, which gives 10𝑦; and 160 minus 120, which is 40. So we’ve eliminated the 𝑥-variable from our equation. We can then solve for 𝑦 by dividing both sides of this equation by 10, giving 𝑦 equals four.

To solve for 𝑥, we substitute this value of 𝑦 into any of our four equations. I’m going to choose equation 1. Doing so gives a straightforward linear equation for 𝑥 which we can solve by subtracting 24 and dividing by four to give 𝑥 equals four. So we have our solution. Both 𝑥 and 𝑦 are equal to four. As always, we should check our solution by substituting our values into any of the other equations. I’ve used equation 2. And it confirms that our solution is correct.

The key step in this question then was to multiply both equations by some number to create the same coefficient of one of the variables. We could then use the method of elimination to eliminate this variable and solve our simultaneous equations.