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Ex 1:

A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations.

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Ex 1:

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  1. A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations. Recall that a line is an infinite set of points that are solutions to a linear equation. The solution of a system of equations is the set of all points that satisfy each equation. x – 3y = –8 (1, 3); 3x + 2y = 9 3x + 2y = 9 x – 3y = –8 3(1) +2(3) –8 9 (1) –3(3) –8 –8  9 9  On the graph of the system of two equations, the solution is the set of points where the lines intersect. A point is a solution to a system of equation if the x- and y-values of the point satisfy both equations. Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. Ex 1: Because the point is a solution for both equations, it is a solution of the system. Substitute 1 for x and 3 for y in each equation.

  2. y= x – 1 y= x – 2 Ex 2: Use a graph and a table to solve the system. Check your answer. 2x – 3y = 3 y + 2 = x Solve each equation for y. On the graph, the lines appear to intersect at the ordered pair (3, 1)

  3. y= x – 1 x y 0 –1 1 2 3 1 y= x – 2 Make a table of values for each equation. Notice that when x = 3, the y-value for both equations is 1. The solution to the system is (3, 1).

  4. The systems of equations in Example 2 have exactly one solution. However, linear systems may also have infinitely many or no solutions. A consistent system is a set of equations or inequalities that has at least one solution, and an inconsistent system will have no solutions. You can classify linear systems by comparing the slopes and y-intercepts of the equations. An independent systemhas equations with different slopes.A dependent system has equations with equal slopes and equal y-intercepts.

  5. y = x– 3 y = x– 3 Ex 3: Classify the system and determine the number of solutions. x = 2y + 6 3x – 6y = 18 Theequations have the same slope and y-intercept and are graphed as the same line. Solve each equation for y. The system is consistent and dependent with infinitely many solutions.

  6. Ex 4: City Park Golf Course charges $20 to rent golf clubs plus $55 per hour for golf cart rental. Sea Vista Golf Course charges $35 to rent clubs plus $45 per hour to rent a cart. For what number of hours is the cost of renting clubs and a cart the same for each course? Step 1 Write an equation for the cost of renting clubs and a cart at each golf course. City Park Golf Course: y = 55x + 20 Sea Vista Golf Course: y = 45x + 35 Because the slopes are different, the system is independent and has exactly one solution.

  7. Use increments of to represent 30 min. When x = , the y-values are both 102.5. The cost of renting clubs and renting a cart for hours is $102.50 at either company. So the cost is the same at each golf course for hours. Step 2 Solve the system by using a table of values. y = 55x + 20 y = 45x + 35

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