Two equations, two letters. Kill one letter and the rest is arithmetic — unless the question never wanted the letters at all.
What is the value of x + y?
5x + 2y = 26
3x − 2y = 6
The solution to the given system of equations is (x, y). What is the value of x + y?
The y-terms are already opposites, so adding the equations gives 8x = 32 and x = 4. Then 2y = 26 − 20 = 6, so y = 3, and x + y = 7. Check both originals: 20 + 6 = 26 ✓ and 12 − 6 = 6 ✓. C is 4, the value of x, and B is 3, the value of y — each is the answer to a question that was not asked, and both are printed because most people who get this wrong get it wrong there. A is 1, the value of x − y, which is what you get from subtracting the equations instead of adding at the very last step.
1
If a letter is already alone, substitute.
Solve the system y = 3x − 5 and 2x + 4y = 22.
equation 1
y = 3x − 5
y stands alone
equation 2
2x + 4y = 22
both letters tangled
2x + 4(3x − 5) = 22
every y in equation 2 becomes the whole bracket
2x + 12x − 20 = 22
distribute the 4 across both terms
14x = 42, x = 3
collect, add 20, divide by 14
y = 3(3) − 5 = 4
back into the simple equation, never the messy one
(x, y) = (3, 4) · check: 2(3) + 4(4) = 22 ✓
Keep the brackets
Writing 2x + 4 · 3x − 5 quietly drops the multiplication on the −5 and produces x = 2714 out of nowhere. The bracket is the reminder that the 4 hits both terms. A coefficient of 1 is just as good as an isolated letter: from x − 3y = 2 you can write x = 3y + 2 in one move and substitute that.
2
If the coefficients already line up, eliminate.
Opposite coefficients — add. Identical coefficients — subtract. Either way one column vanishes and one letter is left.
E15x + 2y = 26
E23x − 2y = 6
5x + 2y = 26
+ 3x − 2y = 6
8x 0y = 32
+2y and −2y add to nothing, which is the whole idea.
8x = 32, x = 4
divide both sides by 8
5(4) + 2y = 26
back into E1
2y = 6, y = 3
and 3(4) − 2(3) = 6 ✓ in E2
When the matching coefficients have the same sign you subtract instead, and every sign on the second line flips. From 4x + 3y = 19 and 4x + y = 9, subtracting gives 2y = 10 — the x column cancels and so does nothing else.
3
If they do not line up, make them line up.
Solve 3x + 4y = 24 and 5x − 2y = 14. Nothing cancels yet — but doubling the second equation fixes that.
E13x + 4y = 24
E25x − 2y = 14× 2
4 and 2 both divide into 4
so only one equation needs scaling
E2 × 2: 10x − 4y = 28
every term, both sides — the 14 doubles too
3x + 4y = 24
+ 10x − 4y = 28
13x 0y = 52
x = 5213 = 4
the calculator is there; use it
3(4) + 4y = 24, 4y = 12, y = 3
check in E2: 20 − 6 = 14 ✓
The right-hand side gets multiplied too
Doubling 5x − 2y = 14 into 10x − 4y = 14 is the single most common slip in elimination. It stops being the same equation, and the answer that comes out satisfies neither original. If you scale, scale everything.
4
Choosing between them takes about three seconds.
Either method solves any system. Look at the shapes and pick the one that makes less work.
What you see
Use
Why
y = … or x = …
substitution
the expression is ready to drop in
a letter with coefficient 1
substitution
isolating it costs one move and no fractions
matching or opposite coefficients
elimination
add or subtract and a column disappears
both equations in Ax + By = C
elimination
they are already stacked in columns
a letter alone → substitutecolumns match → eliminatescale the smaller equation
Isolating a letter whose coefficient is not 1 drags fractions into every line that follows. From 3x + 4y = 24 you can write x = 24 − 4y3, but eliminating avoids that entirely.
5
When the question asks for x + y, you may not need x or y at all.
This is the SAT's favourite move on this topic. Before solving, look at what adding or subtracting the two equations gives you — it is often the exact expression asked for.
E13x + 5y = 14
E25x + 3y = 18
“What is the value of x + y?”
add: 8x + 8y = 32
the x and y coefficients came out equal
8(x + y) = 32
factor the 8 out of the left side
x + y = 4
done, and x and y were never found
subtract instead: 2x − 2y = 4, so x − y = 2
the other half of the same trick
It is not always a plain sum. Sometimes one subtraction produces the whole expression:
E13x + y = 11
E22x + 3y = 15
“What is the value of 2y − x?”
E2 − E1: (2x − 3x) + (3y − y) = 15 − 11
subtract column by column
−x + 2y = 4, so 2y − x = 4
exactly the expression the stem asked for
the actual solution is x = 187, y = 237
ugly on purpose — solving fully is the slow road
2(237) − 187 = 467 − 187 = 287 = 4 ✓
the shortcut and the long way agree
Try one line, then stop trying
Add the equations, then subtract them. If neither gives a multiple of the expression the stem wants, there is no shortcut hiding and you should just solve the system normally — the search is meant to cost one line, not three minutes. Ugly coefficients such as 4x + 9y and 9x + 4y are the strongest hint that a shortcut exists, because nobody sets those numbers by accident.
6
The answer is a crossing point — and Desmos will find it for you.
The opening system drawn in the xy-plane. Two straight lines meet once, and that single point is the solution.
Where the calculator earns its keep
Type both equations into Desmos exactly as printed — it accepts 5x + 2y = 26 without rearranging — and click the crossing point. It reports (4, 3) and you are done. This is the fastest route whenever the coefficients are unpleasant, whenever the solution is a decimal, and whenever you want to see at a glance that two lines are parallel and the system has no solution. It is not faster for the x + y items in step 5: the graph hands you x and y, and you still have to add them.
Answer the question that was asked
Stopping at x is the most expensive habit on this topic. Underline the final quantity — x, y, x + y, 2y − x, xy — before you start, and circle back to it after the algebra. In the opening problem, 4 and 3 were both printed as choices, and both are numbers you genuinely calculate on the way to 7.
✓
Check: both original equations, every time.
the opener5(4) + 2(3) = 26 · 3(4) − 2(3) = 6both true, so x + y = 7 ✓
scaled elimination5(4) − 2(3) = 14the equation that was doubled still works ✓
Remember the flow
Letter alone → substitute · Columns match → eliminate · Otherwise scale · Then answer what was asked
One sum or one difference may finish the question on its own.
Now you try
Practice problem
What is the value of x + y?
4x + 9y = 31
9x + 4y = 34
The solution to the given system of equations is (x, y). What is the value of x + y?
Adding the equations gives 13x + 13y = 65, so 13(x + y) = 65 and x + y = 5 — without ever finding x or y, which are 2.8 and 2.2 and not worth the trouble. D is 65, the sum of the right-hand sides, with the 13 left undivided. B is x and A is y: both come from solving the system all the way down and then answering the wrong question. Coefficients that mirror each other, 4 and 9 against 9 and 4, are the signal that adding will collapse the whole thing in one line.
Hint: you are not asked for x, and you are not asked for y. Add the two equations and look hard at what comes out before you reach for elimination in the ordinary way.
