System of Equations Calculator

by
System of Equations Calculator
2 Equations, 2 Variables (x, y)
Equation 1: a₁x + b₁y = c₁
x +
y =
Equation 2: a₂x + b₂y = c₂
x +
y =
Solution:
Please enter coefficients and click Solve.
function calculateResult(){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 fill in all coefficient fields with valid numbers.');return;}var D=(a1*b2)-(a2*b1);var Dx=(c1*b2)-(c2*b1);var Dy=(a1*c2)-(a2*c1);var stepHtml=";if(D===0){if(Dx===0 && Dy===0){stepHtml='The equations represent the same line.';document.getElementById('finalResult').innerHTML='Infinite Solutions';}else{stepHtml='The lines are parallel and do not intersect.';document.getElementById('finalResult').innerHTML='No Solution';}document.getElementById('stepDetails').innerHTML=stepHtml;return;}var x=Dx/D;var y=Dy/D;if(showSteps){stepHtml='Determinant (D) = ('+a1+'*'+b2+') – ('+a2+'*'+b1+') = '+D+'
';stepHtml+='Dx = ('+c1+'*'+b2+') – ('+c2+'*'+b1+') = '+Dx+'
';stepHtml+='Dy = ('+a1+'*'+c2+') – ('+a2+'*'+c1+') = '+Dy+'
';stepHtml+='x = Dx/D = '+Dx+'/'+D+'
';stepHtml+='y = Dy/D = '+Dy+'/'+D;}document.getElementById('stepDetails').innerHTML=stepHtml;document.getElementById('finalResult').innerHTML='x = '+x.toFixed(4).replace(/\.?0+$/,"")+', y = '+y.toFixed(4).replace(/\.?0+$/,"");}

Using the System of Equations Calculator

The system of equations calculator is a powerful tool designed to solve sets of linear equations with two variables. Whether you are working on algebra homework or solving complex engineering problems, this tool provides instant solutions using Cramer's Rule. By inputting the coefficients of your equations, you can find where two lines intersect on a coordinate plane.

To use this calculator, ensure your equations are in the standard form: ax + by = c.

Coefficients a₁ and a₂
These are the numbers multiplied by the variable x in the first and second equations respectively.
Coefficients b₁ and b₂
These are the numbers multiplied by the variable y in the equations.
Constants c₁ and c₂
These are the numerical values on the right side of the equals sign.

How the Calculation Works

This calculator utilizes Cramer's Rule, which uses determinants to find the values of unknown variables. For a system of two equations:

x = (c₁b₂ – c₂b₁) / (a₁b₂ – a₂b₁)
y = (a₁c₂ – a₂c₁) / (a₁b₂ – a₂b₁)

The denominator in both formulas is the determinant of the coefficient matrix (D). If D equals zero, the system either has no solution (parallel lines) or infinite solutions (the lines are identical).

  • D (Determinant): Measures if a unique solution exists.
  • 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.

Step-by-Step Example

Example Scenario: Solve the following system of equations:
1) 3x + 2y = 16
2) 7x – 5y = 2

Solution Steps:

  1. Identify coefficients: a₁=3, b₁=2, c₁=16; a₂=7, b₂=-5, c₂=2
  2. Calculate D: (3 * -5) – (7 * 2) = -15 – 14 = -29
  3. Calculate Dx: (16 * -5) – (2 * 2) = -80 – 4 = -84
  4. Calculate Dy: (3 * 2) – (7 * 16) = 6 – 112 = -106
  5. Find x: -84 / -29 ≈ 2.8966
  6. Find y: -106 / -29 ≈ 3.6552

Frequently Asked Questions

What if the determinant is zero?

If the determinant D is 0, it means the two equations represent lines with the same slope. If they have different y-intercepts, there is no solution. If they are the exact same line, there are infinitely many solutions.

Can this calculator solve 3×3 systems?

This specific version is optimized for 2×2 systems (two variables). For systems with three variables (x, y, z), you would require a 3×3 matrix determinant calculation, which follows a similar logic but involves more steps.

Does the order of equations matter?

No, the order in which you enter the equations does not change the final result for x and y. However, ensure that each individual equation is consistently formatted with x, then y, then the constant.

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