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 Systems of Equation by Substitution and Elimination

The problems with the graphing method are threefold:

• You need an accurate graph.
• Your graph may not be large enough.
• It’s hard to estimate the solution if the coordinates are not integers.

In this section, we look at two algebraic methods for finding solutions.

• In both of the methods outline below, there are actually three possible outcomes.
• You get a single ordered pair as a solution.

In this case, the solution is the ordered pair you find.

• All variables go away and you get a false statement, such as 0 = 4.

In this case, you have parallel lines, so there is no solution to the system.

• All variables go away and you get a true statement, such as 0 = 0 or 5 = 5.

In this case, you have the same line, so there are infinitely many solutions.

## Substitution

Procedure: (Substitution Method)

0. Choose a variable and an equation.

1. Solve for the chosen variable in the chosen equation.

2. Substitute the expression you found for the selected variable in the OTHER equation.

3. Solve the resulting equation in one variable.

4. Use the answer you found in 3 to find the value of the other variable.

Example:

x + y = 8

2x - 3y = -9

(3, 5).

## Elimination

Procedure: (Elimination Method)

0. Choose a variable.

1. Multiply one or both equations by whatever is necessary to get the coefficients of the selected variable to be the same, but with opposite signs.

2. Add the equations together. (NOTE: This eliminates the selected variable.)

3. Solve the resulting equation in one variable.

4. Use the answer you found in 3 to find the value of the other variable.