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**Solving Systems of Equations**

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Vocabulary System of Equations: a set of equations with the same unknowns Consistent: a system of equations with at least one ordered pair that satisfies both equations. Dependent: a system of equations that has an infinite number of solutions; (lines are the same) Inconsistent: a system of equations with no solution Independent: a system of equations with exactly one solution Solution to a system of equations: a point that makes both equations true

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**Different Methods for Solving Systems**

Elimination Method: adding or subtracting the equations in the systems together so that one of the variables is eliminated; multiplication might be necessary first Substitution Method: Solving one of a pair of equations for one of the variables and substituting that into the other equation Graphing Method: Solving a system by graphing equations on the same coordinate plane and finding the point of intersection.

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**Substitution Method STEPS:**

Solve one of the equations for one of the variables Substitute, or replace, the resulting expression into the other equation for the solved variable Solve the equation for the second variable Substitute the found variable into either of the original equations to find the value of the other variable.

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**How do I write the solution?**

If both values are found, the answer is written as an ordered pair If the resulting solution is a true statement, like 9=9, the system has an infinite number of solutions If the resulting solution is an untrue statement, like 0=9, then the system has no solution

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Example 1 Solve the following system by substitution

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Example 2 Solve the following system using substitution:

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Example 3 Solve the following system by substitution

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**Elimination Method STEPS:**

If needed, multiply each term of the equation by the same number in order to get one pair of variables that are opposite signs but the same coefficient. Add or subtract the two equations to eliminate one of the variables Solve the equation for the second variable Substitute the found value into either of the original equations to find the value of the other variable

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Example 4 Solve the following system using elimination

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Example 5 Solve the following system using elimination:

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Example 6 Solve the following system using elimination

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Applications-Ex 1 Ticket sales to the local gaming convention, PlayerCon, are on the rise! Sales of 2-day tickets brought in $188,100. A total of 6,600 tickets were sold. Adult 2-day tickets cost $36.00 and children’s 2-day tickets cost $ How many of each kind were sold?

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Applications – Ex2 The sales of 3-day tickets to PlayerCon brought in $347,600. The combined cost of one 3-day adult ticket and one 3-day children’s ticket is $90. One-third more adult 3-day tickets were sold than adult 2-day tickets. One-fifth more 3-day children’s tickets were sold than 2-day children’s tickets. What was the cost of 3-day adult tickets and 3-day children’s tickets?

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Warm Up Benito is a waiter. He earns a base salary of $1,500 a month, plus 20% of the price of the meals he serves. Write an equation to predict the amount of money Benito will earn if he serves $350 in meals. Benito earned $1,650 in one month. What was the total price of the meals that Benito served? Create a graph to show the possible amount of money Benito could earn each month.

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**Graphing Method STEPS: Graph both equations on the same plane**

The point of intersection is the solution Special cases: The same line: infinite solutions Parallel lines: no solution

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Example 1 Solve the following system using the graphing method

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Example 2 Solve the system of equations using the graphing method

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Example 3 Solve the following system using the graphing method

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Applications Finn has recently been offered a part-time job as a salesperson at a local cell phone store. He has a choice of two different pay scales. The first option is to receive a base salary of $300 a week plus 10% of the price of the merchandise he sells. The second option is a base salary of $220 a week plus 18% of the price of the merchandise he sells. The average phone sells for $200, but accessories are also included in the merchandise sales. At what amount of sales does Finn receive the same amount of money, regardless of which plan he chooses? Which of the two options should Finn choose and why?

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Solve one of the equations for one of the variables. Isolate one of the variables in one of the equations. Choose whichever seems easiest. Substitute.

Solve one of the equations for one of the variables. Isolate one of the variables in one of the equations. Choose whichever seems easiest. Substitute.

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