Find all solutions to the simultaneous equations 𝑦 plus 𝑥 equals seven and 2𝑥 squared plus 𝑥 plus three 𝑦 equals 21.
To solve simultaneous equations, there’s a couple of different ways. We can solve by graphing them and wherever they intersect will be their solutions, we could use elimination, we could use substitution. Since one of our equations is linear, the 𝑦 plus 𝑥 equals seven, it’s pretty small and it would be easy to solve for 𝑥 or 𝑦, just to isolate it. So we could use substitution because once we would have isolated one of those variables, we can plug it into the other one and solve.
So let’s go ahead and take this equation and subtract 𝑥 from both sides. So we get that 𝑦 is equal to negative 𝑥 plus seven. And we can take that value for 𝑦 and plug that in for 𝑦 into our other equation, and that’s why it’s called substitution. Now, that we’ve substituted this in, this entire equation is in terms of 𝑥; so that’s good, so that means we can solve for 𝑥 now.
Let’s go ahead and distribute the three. Three times negative 𝑥 is negative three 𝑥 and three times seven is 21. And now, we bring down the rest of the equation. So now, let’s combine like terms. We can put the 𝑥s together, which would be negative two 𝑥, bring down the two 𝑥 squared, and bring the 21 on the right side of the equation over to the left side. And they simply cancel, so we are left with two 𝑥 squared minus two 𝑥 equals zero.
So now to solve for 𝑥, we can factor. So first let’s find the greatest common factor, something we can take out of both terms, and that would be two 𝑥. So if we would take out two 𝑥 from two 𝑥 squared, we would have 𝑥 left. And then if we took two 𝑥 out of negative two 𝑥, we would have a negative one left. Now to solve, we said each factor equals zero. So we said two 𝑥 equal to zero and 𝑥 minus one equal to zero. So we need to divide both sides by two for our first equation. And we get 𝑥 equals zero. And on our other equation, we need to add one to both sides. So we get one. So this means 𝑥 can equal zero and it can also equal one.
So what would be the 𝑦-values? So we need to take zero and one and plug it into one of the original equations, so it wouldn’t matter which one. Let’s go ahead and plug it into the linear equation. Now, these two equations are the exact same thing. So our second equation that we kinda have moved 𝑥 over to the right-hand side, it’s equal to the original.
So let’s go ahead and use that second one because 𝑦 is already by itself, and that’s what we’re trying to solve for. So when we plug in 𝑥 equals zero, we get seven for 𝑦. So zero, seven is one of the solutions. And when we plug in one, we get six. So one, six is another solution. Therefore, all of the solutions to simultaneous equations would be zero, seven and one, six.