Systems of equations and graphing systems of linear inequalities
How to find the solution to a system of equations three ways, then how to graph a system of inequalities: boundary lines, shading and the overlap.
A system is two or more equations (or inequalities) that must be true at the same time. The solution to a system of equations is the point where their graphs cross. Graphing systems of linear inequalities works the same way, except the answer is a whole shaded region — every point where all the inequalities hold.
The solution to a system of equations
Worked example
Easy: elimination
Solve and .
Show the working
- Add the equations; cancels.
- Substitute back.
- Check in both: ✓ and ✓.
Answer:
Checked by Sumzo’s math engine
Worked example
Medium: substitution
Solve and .
Show the working
- Substitute into the second equation.
- Solve.
- Check: ✓.
Answer:
Checked by Sumzo’s math engine
Graphically, each linear equation is a line and the solution is where they cross. Parallel lines never cross — no solution; the same line twice gives infinitely many solutions.
How to graph a system of inequalities
- Draw each boundary line as if the inequality were “=”.
- Use a solid line for or (points on the line count) and a dashed line for or .
- Pick a test point not on the line, usually . If it makes the inequality true, shade its side; if not, shade the other side.
- The solution is the region shaded by every inequality — the overlap.
Worked example
Hard: graphing systems of linear inequalities
Graph the system and , and find the corner point.
Show the working
- Boundary 1: , solid. Test : is true, so shade the side containing the origin (below the line).
- Boundary 2: , dashed. Test : is false, so shade the side away from the origin (above the line).
- The corner of the region is where the lines meet.
- The corner lies on the dashed line, so it is not itself a solution; the solutions are the overlap of the two shaded regions.
Answer: The region below and above , to the right of their crossing point .
Checked by Sumzo’s math engine
Common mistakes
- Using a solid line for a strict inequality (, ).
- Shading the wrong side — always test a point.
- Forgetting to flip the sign when multiplying or dividing an inequality by a negative number.
- Giving only as the solution of a system — the answer is a pair .
Frequently asked questions
How do you graph a system of inequalities?
Graph each boundary line (solid for ≤ or ≥, dashed for < or >), shade the side that makes each inequality true using a test point, and the solution is where all the shading overlaps.
What is the solution to a system of equations?
The pair of values that makes every equation true at once — on a graph, the point where the lines cross.
When does a system have no solution?
When the lines are parallel (same slope, different intercepts). For inequalities, when the shaded regions don’t overlap.