Systems of Equations Calculator

by
systems of equations calculator
Linear System: 2 Equations, 2 Variables (x, y)
Enter Coefficients for Standard Form: ax + by = c
x +
y =
x +
y =
Solution:
Enter coefficients to see the result.
function calculateSystem(){var a1=parseFloat(document.getElementById('a1').value);var b1=parseFloat(document.getElementById('b1').value);var c1=parseFloat(document.getElementById('c1').value);var a2=parseFloat(document.getElementById('a2').value);var b2=parseFloat(document.getElementById('b2').value);var c2=parseFloat(document.getElementById('c2').value);var showSteps=document.getElementById('steps').checked;if(isNaN(a1)||isNaN(b1)||isNaN(c1)||isNaN(a2)||isNaN(b2)||isNaN(c2)){alert('Please enter valid numeric coefficients in all fields.');return;}var D=(a1*b2)-(a2*b1);var Dx=(c1*b2)-(c2*b1);var Dy=(a1*c2)-(a2*c1);var output="";if(D!==0){var x=Dx/D;var y=Dy/D;output="
x = "+x.toFixed(4).replace(/\.?0+$/,"")+", y = "+y.toFixed(4).replace(/\.?0+$/,"")+"
";if(showSteps){output+="
";output+="Determinant (D) = ("+a1+"×"+b2+") – ("+a2+"×"+b1+") = "+D+"
";output+="Dx = ("+c1+"×"+b2+") – ("+c2+"×"+b1+") = "+Dx+"
";output+="Dy = ("+a1+"×"+c2+") – ("+a2+"×"+c1+") = "+Dy+"
";output+="x = Dx / D = "+Dx+" / "+D+" = "+x.toFixed(4)+"
";output+="y = Dy / D = "+Dy+" / "+D+" = "+y.toFixed(4);output+="
";}}else{if(Dx===0&&Dy===0){output="
Infinite Solutions
The equations represent the same line (dependent system).
";}else{output="
No Solution
The lines are parallel and never intersect (inconsistent system).
";}}document.getElementById('answer').innerHTML=output;}

Calculator Use

The systems of equations calculator is a powerful tool designed to solve sets of linear equations with two variables. Whether you are a student working on algebra homework or a professional solving engineering problems, this tool provides instant solutions using the standard form (ax + by = c).

To use the calculator, simply enter the coefficients for your two equations. The calculator identifies the values for x and y where the two lines intersect on a coordinate plane.

Coefficients (a1, b1, a2, b2)
These are the numbers multiplied by the variables x and y. For example, in the equation 3x + 4y = 10, 3 is 'a' and 4 is 'b'.
Constants (c1, c2)
These are the numbers on the right side of the equals sign in standard form.
Show Determinants
Checking this box will display the mathematical steps using Cramer's Rule, showing you exactly how the final result was derived.

How It Works: Cramer's Rule

This systems of equations calculator utilizes Cramer's Rule, a mathematical theorem that uses determinants to solve systems of linear equations. For a 2×2 system, the process follows these formulas:

D = (a1 * b2) – (a2 * b1)
Dx = (c1 * b2) – (c2 * b1)
Dy = (a1 * c2) – (a2 * c1)

  • D (Main Determinant): If D is not zero, the system has a unique solution.
  • Dx: The determinant formed by replacing the x-column with the constant column.
  • Dy: The determinant formed by replacing the y-column with the constant column.
  • Solution: x = Dx / D and y = Dy / D.

Calculation Example

Example: Solve the following system of equations:
1) 2x + 3y = 8
2) 4x – y = 2

Step-by-step solution:

  1. Identify coefficients: a1=2, b1=3, c1=8 and a2=4, b2=-1, c2=2
  2. Calculate D: (2 * -1) – (4 * 3) = -2 – 12 = -14
  3. Calculate Dx: (8 * -1) – (2 * 3) = -8 – 6 = -14
  4. Calculate Dy: (2 * 2) – (4 * 8) = 4 – 32 = -28
  5. Solve for x: -14 / -14 = 1
  6. Solve for y: -28 / -14 = 2
  7. Result: x = 1, y = 2

Common Questions

What happens if the determinant is zero?

If the main determinant (D) is zero, the lines are parallel. If the other determinants (Dx and Dy) are also zero, the lines are identical, meaning there are infinite solutions. If Dx or Dy are non-zero, the lines are parallel and distinct, meaning there is no solution.

Can this solve non-linear equations?

No, this specific systems of equations calculator is designed for linear equations where variables are to the first power. Solving non-linear systems (like those containing x²) requires more complex methods like substitution or Newton's method.

What is Standard Form?

Standard form for a linear equation is Ax + By = C. If your equation is in slope-intercept form (y = mx + b), you must rearrange it. For example, y = 2x + 5 becomes -2x + y = 5.

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