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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

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.

5. Write your answer as an ordered pair.

Example:

x + y = 8

2x - 3y = -9

Answer:

(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.

5. Write your answer as an ordered pair.

Example:

x + y = 8

3x - y = 0

Answer:

(2, 6).

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