Systems of Equations Calculator
A system’s solution is the point that satisfies every equation: 2x + 3y = 12 and x − y = 1 meet at x = 3, y = 2.
Example: 2x + 3y = 12 is “2 3 12”; x − y = 1 is “1 -1 1”.
x = 3, y = 2
- x (decimal)
- 3
- y (decimal)
- 2
- Determinant of coefficients
- -5
Step-by-step solution
- 2x + 3y = 12
- x − y = 1
- Cramer’s rule: each unknown = (determinant with that column replaced by the constants) ÷ -5
- x = 3, y = 2
Systems of Equations Formula
a₁x + b₁y = c₁, a₂x + b₂y = c₂
solved by elimination / Cramer’s rule; determinant 0 means no unique solution
Worked Example
The system 2x + 3y = 12 and x − y = 1:
- 2x + 3y = 12
- x − y = 1
- Cramer’s rule: each unknown = (determinant with that column replaced by the constants) ÷ -5
- x = 3, y = 2
y (decimal): 2 · Determinant of coefficients: -5
Test-day tip
With the digital SAT’s graphing calculator, type both equations exactly as given and click the intersection point. For “no solution” questions, check whether the lines are parallel instead of solving.
More Algebra Calculators
-
Slope-Intercept Form Calculator
y = mx + b
m slope, b y-intercept = y₁ − m·x₁
Example: The line through (1, 2) and (4, 8): Slope-intercept form: y = 2x.
-
Linear Inequality Calculator
ax + b < c → x < (c − b)/a
the inequality sign reverses when dividing by a negative a
Example: The inequality −3x + 4 < 10: Solution = x > -2.
-
Slope Calculator
m = (y₂ − y₁) ÷ (x₂ − x₁)
rise over run between points (x₁, y₁) and (x₂, y₂)
Example: The line through (1, 2) and (4, 8): Slope m = 2.
-
Point-Slope Form Calculator
y − y₁ = m(x − x₁)
point (x₁, y₁), slope m
Example: The line through (1, 2) with slope 3: Point-slope form: y − 2 = 3(x − 1).
Frequently asked questions
How can I tell if a system has no solution?
For two lines, no solution means equal slopes and different intercepts (parallel lines). Infinitely many solutions means the equations describe the same line.
For what value of k does a system have no solution?
Set the ratios a₁/a₂ = b₁/b₂ but ≠ c₁/c₂ and solve for k. This is a common SAT question type.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Systems of Equations Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/systems-of-equations-calculator/
IEEE
Scientific Converters, "Systems of Equations Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/systems-of-equations-calculator/