Desmos Tricks

System Solver

Solve systems of linear equations instantly. No substitution, no elimination. Just the gray dots.

📝 SAT Problem

\[ 2x + 3y = 12, \quad y = x - 1 \]

The system of equations above has solution \((x, y)\). What is the value of \(5x\)?

(A) 2
(B) 3
(C) 15
(D) 25

Step 1: Type Equations for Digital SAT System of Equations Trick

Don't rearrange for \(y\). Just type them into two separate lines in Desmos.

2x + 3y = 12
y = x - 1

Step 2: Find Intersection to Solve SAT Systems Promptly

Click the point where the lines cross. Desmos displays a gray dot at (3, 2).

Digital SAT system of equations Desmos trick displayed on calculator

The question asks for \(5x\). Since \(x = 3\), the answer is \(5(3) = 15\).

Final Answer:

15 (Answer choice C)

🚫 The "No Solution" Trap

When questions ask for "No Solution", they mean Parallel Lines (Same Slope).

📝 SAT Problem

\[ kx - 3y = 4, \quad 4x - 5y = 7 \]

For what value of \(k\) does the system of equations above have no solution?

(A) 1.2
(B) 2.4
(C) 3.6
(D) 4.8

Step 1: Test Options

Substitute each option for \(k\) and look for perfectly parallel lines that never cross.

(A) \( k = 1.2 \)
k=1.2 graph
(B) \( k = 2.4 \) ✅
k=2.4 graph
(C) \( k = 3.6 \)
k=3.6 graph
(D) \( k = 4.8 \)
k=4.8 graph

Only Option B produces parallel lines, meaning there is "No Solution" state.

Final Answer:

2.4 (Answer choice B)

🎨 The Shading Zone

For systems of inequalities, the answer is in the double-shaded region.

📝 SAT Problem

\[ y < 2x + 1, \quad y> -x + 3 \]

Which point \((x, y)\) is a solution to the system of inequalities above?

(A) (0, 0)
(B) (2, 0)
(C) (0, 5)
(D) (4, 2)

Step 1: Graph & Spot the Dark Zone

Type both inequalities. Look for where the blue and red shadings overlap (the darkest region).

Desmos inequality shading showing solution region

Step 2: Check the Points

Type the point \((4, 2)\). Since it lands inside the double-shaded region, it is the correct answer.

Final Answer:

(4, 2) (Answer choice D)

⚔️ Mastered the Intersection Hack?

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