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Solve The Following System of Equations Algebraically Calculator

Reviewed by Calculator Editorial Team

This guide explains how to solve systems of equations algebraically using substitution, elimination, and matrix methods. The calculator on this page provides step-by-step solutions for any system you input.

Introduction to Solving Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving a system means finding values for the variables that satisfy all equations simultaneously. There are several algebraic methods to solve systems:

  • Substitution Method: Solve one equation for one variable and substitute into the other equation.
  • Elimination Method: Add or subtract equations to eliminate one variable.
  • Matrix Method: Use matrices and determinants to solve larger systems.

This guide focuses on the algebraic methods, which are fundamental for understanding more advanced techniques.

Solving Methods

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Here's a step-by-step process:

  1. Choose one equation and solve for one variable in terms of the other.
  2. Substitute this expression into the second equation.
  3. Solve the resulting one-variable equation.
  4. Substitute the found value back into one of the original equations to find the other variable.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable. Here's how it works:

  1. Write both equations in standard form (Ax + By = C).
  2. Make the coefficients of one variable the same by multiplying the equations.
  3. Add or subtract the equations to eliminate one variable.
  4. Solve the resulting one-variable equation.
  5. Substitute the found value back into one of the original equations to find the other variable.

Key Formulas

For a system of two linear equations:

1. Equation 1: a₁x + b₁y = c₁

2. Equation 2: a₂x + b₂y = c₂

The solution (x, y) can be found using substitution or elimination.

Worked Examples

Example 1: Substitution Method

Solve the system:

2x + 3y = 8

x - y = 1

Solution:

  1. From the second equation: x = y + 1
  2. Substitute into the first equation: 2(y + 1) + 3y = 8 → 2y + 2 + 3y = 8 → 5y = 6 → y = 6/5
  3. Substitute y back: x = (6/5) + 1 = 11/5
  4. Solution: (11/5, 6/5)

Example 2: Elimination Method

Solve the system:

3x + 2y = 13

x - y = 2

Solution:

  1. Multiply the second equation by 2: 2x - 2y = 4
  2. Add to the first equation: (3x + 2y) + (2x - 2y) = 13 + 4 → 5x = 17 → x = 17/5
  3. Substitute into the second equation: (17/5) - y = 2 → y = 17/5 - 2 = 7/5
  4. Solution: (17/5, 7/5)

Note: Always check your solutions by substituting them back into the original equations to ensure they satisfy both.

FAQ

What is the difference between substitution and elimination?
The substitution method involves solving for one variable and substituting into another equation. The elimination method involves adding or subtracting equations to eliminate one variable.
When should I use substitution versus elimination?
Use substitution when one equation is easily solvable for one variable. Use elimination when the coefficients of one variable are easily made equal.
What if a system has no solution?
A system has no solution if the equations represent parallel lines (same slope, different intercepts). This occurs when the determinant of the coefficient matrix is zero.
What if a system has infinitely many solutions?
A system has infinitely many solutions if the equations represent the same line (same slope and intercept). This occurs when the determinant of the coefficient matrix is zero and the equations are dependent.