Solve the simultaneous equations 𝑦 plus four 𝑥 equals negative eight and 𝑦 equals five 𝑥 plus 10.
When we’re solving two simultaneous equations, what we’re trying to figure out is where both of these equations would be equal to the same amounts. And if we were to graph them, this would be the place where these two lines intersect.
Our first equation, 𝑦 plus four 𝑥 equals negative eight, can be solved if we substitute 𝑦 equals five 𝑥 plus 10. So we’ll substitute five 𝑥 plus 10 in place of 𝑦 in our first equation. This means we now have five 𝑥 plus 10 plus four 𝑥 equals negative eight.
In this equation, we’ll need to solve for 𝑥. Five 𝑥 plus four 𝑥 equals nine 𝑥. Bring everything else down from our equation. To isolate 𝑥, we’ll need to subtract 10 from both sides of our equation, leaving us with nine 𝑥 equal to negative 18.
On the left side, 𝑥 is being multiplied by nine. So to get rid of that, we can divide by nine on both sides of our equation. Nine 𝑥 divided by nine equals 𝑥; negative 18 divided by nine equals negative two.
Now that we know what our 𝑥 is, negative two, we can plug in our 𝑥 value into this second equation. Okay we take 𝑥 equals negative two and we plug that in for 𝑥. And now we have five times negative two plus 10 equals 𝑦. Five times negative two equals negative 10; negative 10 plus 10 equals zero, which tells us that 𝑦 equals zero.
If you wanna check the solution, we can plug in our 𝑥 and 𝑦 values into both of these equations to make sure they’re true. Our first equation, 𝑦 plus four 𝑥 equals negative eight, we plug in our zero and our negative two, and we have zero plus four times negative two equal negative eight, and that is true. Four times negative two equals negative eight; negative eight equals negative eight.
And for our second equation, 𝑦 equals five 𝑥 plus 10; zero equals five times negative two plus 10. Does zero equal negative 10 plus 10? It does, so we know that our 𝑥 is negative two and our 𝑦 equals zero.