Algebra Tutorials! Home Solving Quadratic Equations by Completing the Square Graphing Logarithmic Functions Division Property of Exponents Adding and Subtracting Rational Expressions With Like Denominators Rationalizing the Denominator Multiplying Special Polynomials Functions Solving Linear Systems of Equations by Elimination Solving Systems of Equation by Substitution and Elimination Polynomial Equations Solving Linear Systems of Equations by Graphing Quadratic Functions Solving Proportions Parallel and Perpendicular Lines Simplifying Square Roots Simplifying Fractions Adding and Subtracting Fractions Adding and Subtracting Fractions Solving Linear Equations Inequalities in one Variable Recognizing Polynomial Equations from their Graphs Scientific Notation Factoring a Sum or Difference of Two Cubes Solving Nonlinear Equations by Substitution Solving Systems of Linear Inequalities Arithmetics with Decimals Finding the Equation of an Inverse Function Plotting Points in the Coordinate Plane The Product of the Roots of a Quadratic Powers Solving Quadratic Equations by Completing the Square
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# Solving Linear Systems of Equations by Elimination

Use elimination to find the solution of this system. Solution

To make the equations easier to work with:

 â€¢ Clear the fractions in the first equation by multiplying both sides by 6, the LCD of the fractions. â€¢ Clear the decimals from the second equation by multiplying both sides by 10. 10(0.3x + 0.2y = -1) → 3x + 2y = -10 To make the x-coefficients opposites, multiply the transformed second equation by -1. Add the equations. Both variables are eliminated. The result is the false statement 0 = 31.

When the result is a false statement, the graphs of the equations never intersect. The graph confirms that the lines are parallel and have no points in common.

This system has no solution because the lines never intersect.

The system is inconsistent. (It has no solution.)

The equations of the system are independent. (Their graphs are not identical.) Note â€” Solving a Linear System: Special Cases

When using either substitution or elimination, if both variables are eliminated there are two possible outcomes:

â€¢ If the resulting equation is an identity, such as 5 = 5, then the lines coincide.

The system has infinitely many solutions.

The solutions may be stated as the set of all points on the line.

â€¢ If the resulting equation is a false statement, such as 0 = 4, then the lines are parallel and never intersect.

The system has no solution.

The system is inconsistent and the equations are independent.