#13 · x + y without x or y.
The question. The solution to this system of equations is (x, y). What is the value of x + y?
The slow way. Eliminate x to find y = −2/3, then plug that back in to find x = 11/3, and add: 11/3 + (−2/3) = 3. It works, but it's a lot of steps, with fractions and negative signs along the way: easy to slip on, and slow.
Add the equations. The question asks for x + y, not for x and y, so look for a way to get there directly. Add the two equations:
Now x and y both have 11 in front, so the left side is 11(x + y). Divide both sides by 11:
So x + y = 3. You never needed x or y on their own.
Desmos works too: add where the lines cross
Graph both equations and click where the lines cross. Desmos shows the point as about (3.66667, −0.66667): rounded decimals of 11/3 and −2/3. They add to about 3. Here, adding the equations gets you there directly, and gives the exact value: x + y = 3.
The same habit on other questions
When a systems question asks for an expression, like x − y or 2x + y, instead of x or y, check whether adding, subtracting, or scaling the equations gets you to it in fewer steps. If nothing lines up, solving for each variable still works.
The takeaway: don't solve more than you need. Here, adding the equations gives 11(x + y) = 33, so x + y = 3.