Resources / SAT / Desmos / Systems & Inequalities

Systems & Inequalities

Find intersections of two equations in one click. Shade inequality regions automatically, useful for "which of the following points satisfies..." questions.

Systems of equations

  1. Type each equation on its own line.
  2. Intersection(s) appear as colored dots.
  3. Click any dot for exact coordinates.
Example: Solve $\begin{cases} 2x + 3y = 12 \\ x - y = 1 \end{cases}$.
  1. Line 1: 2x+3y=12.
  2. Line 2: x-y=1.
  3. Click the intersection. Label: $(3, 2)$.

Answer: $x = 3$, $y = 2$. Zero substitution needed.

Nonlinear systems

Problem: How many solutions does the system $\begin{cases} y = x^2 - 4 \\ y = 2x + 1 \end{cases}$ have?
  1. Type y=x^2-4 and y=2x+1.
  2. Count the intersection dots: two.
  3. Click each for exact points: $(-1.449, -1.898)$ and $(3.449, 7.898)$.

Answer: $2$ solutions.

Linear inequalities (automatic shading)

  1. Type y<=2x+3. Desmos shades the region below (and on) the line.
  2. Use < and > for strict inequalities (dashed boundary).
  3. Type a second inequality on a new line. The overlap region, where both shadings intersect, is the solution set.

Worked example: "which point satisfies..."

Problem: Which ordered pair is in the solution set of $\begin{cases} y > 2x - 1 \\ x + y \le 6 \end{cases}$?
A) $(1, 3)$   B) $(2, 3)$   C) $(4, 2)$   D) $(5, 2)$
  1. Type y>2x-1 and x+y<=6.
  2. Type all four candidate points: (1,3), (2,3), (4,2), (5,2).
  3. Look at which dots fall in the overlap of the two shaded regions.
  4. Only $(1, 3)$ sits in both. Answer: A.

Algebra path: plug each point into both inequalities and check. ~60 seconds. Desmos: 15.

Worked example: feasible region corner

Problem: The solution set of $y \ge 0$, $x \ge 0$, $x + y \le 10$, and $2x + y \le 16$ forms a polygon. What is the maximum value of $x + 2y$ on this region?
  1. Type all four inequalities. Desmos shades the feasible region.
  2. Click each corner of the polygon to read coordinates, $(0, 0)$, $(0, 10)$, $(6, 4)$, $(8, 0)$.
  3. Evaluate $x + 2y$ at each: $0, 20, 14, 8$. Max = $20$.

Answer: $20$ at $(0, 10)$.

Common traps

Watch for these gotchas
  • Answer is not the intersection. Read the question, sometimes the answer is just $x$, sometimes it's the sum $x + y$, sometimes it's a value of a function at the intersection.
  • No intersection on screen. Zoom out (wheel or pinch). If lines are parallel, no intersection means no solution.
  • Shading gets busy. With 3+ inequalities, temporarily hide layers (click the colored circle to toggle) to identify the feasible region.
  • Strict vs. inclusive. $\le$ includes the boundary (solid line); $<$ does not (dashed). Matters for points on the boundary.

Try it yourself

Practice problem
Find the intersection of $y = 3x - 2$ and $y = -x + 6$. Then determine which of these points lies in the region where $y \ge 3x - 2$ AND $y \le -x + 6$:
A) $(1, 4)$   B) $(2, 2)$   C) $(3, 2)$   D) $(4, 1)$
Show answer

Intersection: Type y=3x-2 and y=-x+6. Click the crossing point, $(2, 4)$.

Region test: Type y>=3x-2 and y<=-x+6. Type each candidate point. Only $(1, 4)$ falls in both shaded regions. Answer: A.

Next: Statistics Shortcuts →
Mean, median, standard deviation directly from a table.
Practice Systems
Drill system-of-equations questions from the Algebra domain.