6-3 and 6-4 Solving Systems Using Elimination; Applications of Systems

Quick Review

You can add or subtract equations in a system to eliminate a variable. Before you add or subtract, you may have to multiply one or both equations by a constant to make eliminating a variable possible.

Example

What is the solution of the system?

3 x + 2 y = 41 5 x − 3 y = 24 table with 2 rows and 1 column , row1 column 1 , 3 bold italic x plus 2 bold italic y equals 41 , row2 column 1 , 5 bold italic x minus 3 bold italic y equals 24 , end table

3 x + 2 y = 41 5 x − 3 y = 24 → Multiply by  3. → Multiply by  2. 9 x + 6 y = 123 10 x − 6 y = 48 _     19 x + 0 = 171 x = 9 table with 2 rows and 3 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 3 x plus 2 y , column 2 equals 41 , row2 column 1 , 5 x minus 3 y , column 2 equals 24 , end table , column 2 table with 2 rows and 1 column , row1 column 1 , modified rightwards arrow with cap multiplyby . 3. above , row2 column 1 , modified rightwards arrow with cap multiplyby . 2. above , end table , column 3 modified table with 2 rows and 2 columns , row1 column 1 , 9 x plus 6 y , column 2 equals 123 , row2 column 1 , 10 x minus 6 y , column 2 equals 48 , end table with under bar below , row2 column 1 , , column 2 , column 3 table with 2 rows and 2 columns , row1 column 1 , 19 x plus 0 , column 2 equals 171 , row2 column 1 , x , column 2 equals 9 , end table , end table

3 x + 2 y = 41 Write the first equation . 3 ( 9 ) + 2 y = 41 Substitute  9  for  x . y = 7 Solve for  y . table with 3 rows and 3 columns , row1 column 1 , 3 x plus 2 y , column 2 equals 41 , column 3 cap writethefirstequation . . , row2 column 1 , 3 , open 9 close , plus 2 y , column 2 equals 41 , column 3 cap substitute . 9 , for , x . , row3 column 1 , y , column 2 equals 7 , column 3 cap solvefor . y . , end table

The solution is (9, 7).

Exercises

Solve each system using elimination. Tell whether the system has one solution, infinitely many solutions, or no solution.

19. x + 2 y = 23 5 x + 10 y = 55 table with 2 rows and 1 column , row1 column 1 , x plus 2 y equals 23 , row2 column 1 , 5 x plus 10 y equals 55 , end table
20. 7 x + y = 6 5 x + 3 y = 34 table with 2 rows and 1 column , row1 column 1 , 7 x plus y equals 6 , row2 column 1 , 5 x plus 3 y equals 34 , end table
21. 5 x + 4 y = − 83 3 x − 3 y = − 12 table with 2 rows and 1 column , row1 column 1 , 5 x plus 4 y equals negative 83 , row2 column 1 , 3 x minus 3 y equals negative 12 , end table
22. 9 x + 1 2 y = 51 7 x + 1 3 y = 39 table with 2 rows and 2 columns , row1 column 1 , 9 x plus , 1 half , y , column 2 equals 51 , row2 column 1 , 7 x plus , 1 third , y , column 2 equals 39 , end table
23. 4 x + y = 21 − 2 x + 6 y = 9 table with 2 rows and 1 column , row1 column 1 , 4 x plus y equals 21 , row2 column 1 , negative 2 x plus 6 y equals 9 , end table
24. y = 3 x − 27 x − 1 3 y = 9 table with 2 rows and 1 column , row1 column 1 , y equals 3 x minus 27 , row2 column 1 , x minus , 1 third , y equals 9 , end table
25. Flower Arranging It takes a florist 3 h 15 min to make 3 small centerpieces and 3 large centerpieces. It takes 6 h 20 min to make 4 small centerpieces and 7 large centerpieces. How long does it take to make each small centerpiece and each large centerpiece? Write and solve a system of equations to find your answer.

6-5 and 6-6 Linear Inequalities and Systems of Inequalities

Quick Review

A linear inequality describes a region of the coordinate plane with a boundary line. Two or more inequalities form a system of inequalities. The system's solutions lie where the graphs of the inequalities overlap.

Example

What is the graph of the system?

y > 2 x − 4 y ≤ − x + 2 table with 2 rows and 1 column , row1 column 1 , bold italic y greater than 2 bold italic x minus 4 , row2 column 1 , bold italic y less than or equal to negative bold italic x plus 2 , end table

Graph the boundary lines y = 2 x − 4 y equals 2 x minus 4  and y = − x + 2 . y equals negative x plus 2 .

For y > 2 x − 4 , y greater than 2 x minus 4 . comma  use a dashed boundary line and shade above it. For y ≤ − x + 2 , y less than or equal to negative x plus 2 . comma  use a solid boundary line and shade below. The green region of overlap contains the system's solutions.

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Exercises

Solve each system of inequalities by graphing.

26. y ≥ x + 4 y < 2 x − 1 table with 2 rows and 1 column , row1 column 1 , y greater than or equal to x plus 4 , row2 column 1 , y less than 2 x minus 1 , end table
27. 4 y < − 3 x y < − 3 4 x table with 2 rows and 1 column , row1 column 1 , 4 y less than negative 3 x , row2 column 1 , y less than negative , 3 fourths , x , end table
28. 2 x − y > 0 3 x + 2 y ≤ − 14 table with 2 rows and 1 column , row1 column 1 , 2 x minus y greater than 0 , row2 column 1 , 3 x plus 2 y less than or equal to negative 14 , end table
29. x + 0.5 y ≥ 5.5 0.5 x + y < 6.5 table with 2 rows and 1 column , row1 column 1 , x plus 0.5 y greater than or equal to 5.5 , row2 column 1 , 0.5 x plus y less than 6.5 , end table
30. y < 10 x y > x − 5 table with 2 rows and 1 column , row1 column 1 , y less than 10 x , row2 column 1 , y greater than x minus 5 , end table
31. 4 x + 4 > 2 y 3 x − 4 y ≥ 1 table with 2 rows and 1 column , row1 column 1 , 4 x plus 4 greater than 2 y , row2 column 1 , 3 x minus 4 y greater than or equal to 1 , end table
32. Downloads You have 60 megabytes (MB) of space left on your portable media player. You can choose to download song files that use 3.5 MB or video files that use 8 MB. You want to download at least 12 files. What is a graph showing the numbers of song and video files you can download?

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