SAT Systems of Equations: The Laziest Trick

When they ask for 11x, don't solve for x. Just add the equations.

The Problem

You see this on the SAT:

5x - 3y = 10
6x + 3y = 23

Question: What is the value of 11x?

🚫 The Hard Way (90 Seconds)

Most students try to solve for x step-by-step:

  1. Use elimination to solve for x
  2. Plug x back in to find y
  3. Finally, multiply x by 11

This takes 90+ seconds and involves 3 separate steps. But the question didn't ask for x OR y—it asked for 11x.

✅ The Practix Shortcut (5 Seconds)

Don't solve. Just combine.

Look at what happens when you add the two equations together:

   5x - 3y = 10
+  6x + 3y = 23
_______________
  11x +  0  = 33

Notice:

  • 5x + 6x = 11x (exactly what the question asks for!)
  • -3y + 3y = 0 (the y terms cancel)
  • 10 + 23 = 33

Answer: 11x = 33

You're done. You never solved for x. You never solved for y. You just added two lines of algebra.

When to Use This Trick

This works ANYTIME the question asks for a combination of variables instead of individual values. Examples:

  • "What is 2x + y?"
  • "What is x - 3y?"
  • "What is 4x?"

Strategy: Look for ways to combine the equations (add, subtract, multiply) to create the exact expression they're asking for.

Systems Solver

Enter your system and see what you get by adding/subtracting.

Equation 1: 5x - 3y = 10
Equation 2: 6x + 3y = 23

Result of Adding Equations
11x = 33

🎓 The Proof: Why can we just add equations?

Adding two equations together might feel like "cheating," but it is actually a rigorous property of equality. Here is the logic:

1. Start with two true statements (equations):

$LHS_1 = RHS_1$   and   $LHS_2 = RHS_2$

2. According to the Addition Property of Equality, you can add the same value to both sides of an equation without changing its truth. Let's add $LHS_2$ to both sides of the first equation:

$LHS_1 + LHS_2 = RHS_1 + LHS_2$

3. Since our second equation tells us that $LHS_2 = RHS_2$, we can substitute the right side:

$LHS_1 + LHS_2 = RHS_1 + RHS_2$

This proves that if you have two balanced scales, and you put the contents of one onto the other, the resulting scale must stay balanced. This is why you can instantly combine variables to find $11x$ or $x+y$ on the SAT.

Algebra is the art of maintaining balance. — Practix Team

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