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Example 2

Solve the system of equations by graphing. x – 2 y = 0 x + y = 6. Solve by Graphing. Write each equation in slope-intercept form. The graphs appear to intersect at (4, 2). Example 2. Check Substitute the coordinates into each equation. ?. ?.

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Example 2

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  1. Solve the system of equations by graphing. x – 2y = 0x + y = 6 Solve by Graphing Write each equation in slope-intercept form. The graphs appear to intersect at (4, 2). Example 2

  2. CheckSubstitute the coordinates into each equation. ? ? 4 – 2(2) = 0 4 + 2 = 6 Replace x with 4 and y with 2. Solve by Graphing x – 2y = 0 x + y = 6 Original equations 0 = 0 6 = 6 Simplify. Answer: The solution of the system is (4, 2). Example 2

  3. A. C. B. D. Which graph shows the solution to the system of equations below?x + 3y = 7x – y = 3 Example 2

  4. A. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.x – y = 5x + 2y = –4 Classify Systems Write each equation in slope-intercept form. Example 3

  5. Answer: Classify Systems The graphs of the equations intersect at (2, –3). Since there is one solution to this system, this system is consistent and independent. Example 3

  6. B. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.9x – 6y = –66x – 4y = –4 Classify Systems Write each equation in slope-intercept form. Since the equations are equivalent, their graphs are the same line. Example 3

  7. Answer: Classify Systems Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent. Example 3

  8. C. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.15x – 6y = 05x – 2y = 10 Classify Systems Write each equation in slope-intercept form. Example 3

  9. Answer: Classify Systems The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent. Example 3

  10. D. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.f(x) = –0.5x + 2g(x) = –0.5x + 2h(x) = 0.5x + 2 Classify Systems Example 3

  11. Answer: Classify Systems f(x) and g(x) are consistent and dependent. f(x) and h(x) are consistent and independent. g(x) and h(x) are consistent and independent. Example 3

  12. A. Graph the system of equations below. What type of system of equations is shown? x + y = 52x = y – 5 A. consistent and independent B. consistent and dependent C. consistent D. none of the above Example 3

  13. B. Graph the system of equations below. What type of system of equations is shown? x + y = 32x = –2y + 6 A. consistent and independent B. consistent and dependent C. inconsistent D. none of the above Example 3

  14. C. Graph the system of equations below. What type of system of equations is shown? y = 3x + 2–6x + 2y = 10 A. consistent and independent B. consistent and dependent C. inconsistent D. none of the above Example 3

  15. Concept

  16. Concept

  17. FURNITURE Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13,930 for the 48 outdoor chairs sold. How many of each chair were sold? Use the Substitution Method Understand You are asked to find the number of each type of chair sold. Example 4

  18. Define variables and write the system of equations. Let x represent the number of rocking chairs sold and y represent the number of Adirondack chairs sold. Use the Substitution Method Plan x + y = 48 The total number of chairs sold was 48. 265x + 320y = 13,930The total amount earned was $13,930. Example 4

  19. Solve one of the equations for one of the variables in terms of the other. Since the coefficient of x is 1, solve the first equation for x in terms of y. Use the Substitution Method x + y = 48 First equation x = 48 – y Subtract y from each side. Example 4

  20. SolveSubstitute 48 – y for x in the second equation. Use the Substitution Method 265x + 320y = 13,930 Second equation 265(48 – y) + 320y = 13,930Substitute 48 – y for x. 12,720 – 265y + 320y = 13,930 Distributive Property 55y = 1210 Simplify. y = 22 Divide each side by 55. Example 4

  21. Now find the value of x. Substitute the value for y into either equation. Use the Substitution Method x + y = 48 First equation x + 22 = 48Replace y with 22. x = 26 Subtract 22 from each side. Answer:They sold 26 rocking chairs and 22 Adirondack chairs. Example 4

  22. Use the Substitution Method CheckYou can use a graphing calculator to check this solution. Example 4

  23. AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24.50 for adults and $16.50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold? A. 210 adult; 120 children B. 120 adult; 210 children C. 300 children; 30 adult D. 300 children; 30 adult Example 4

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  25. Use the elimination method to solve the system of equations. x + 2y = 10x + y = 6 Solve by Using Elimination In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated. x + 2y = 10 (–)x + y = 6 y = 4 Subtract the equations. Example 5

  26. Now find x by substituting 4 for y in either original equation. Solve by Using Elimination x + y = 6 Second equation x + 4 = 6 Replace y with 4. x = 2 Subtract 4 from each side. Answer: The solution is (2, 4). Example 5

  27. Use the elimination method to solve the system of equations. What is the solution to the system?x + 3y = 5x + 5y = –3 A. (2, –1) B. (17, –4) C. (2, 1) D. no solution Example 5

  28. Read the Test ItemYou are given a system of two linear equations and are asked to find the solution. A. (2, 3) B. (6, 0) C. (0, 5.5) D. (3, 2) No Solution and Infinite Solutions Solve the system of equations.2x + 3y = 125x – 2y = 11 Example 6

  29. 2x + 3y = 12 4x + 6y = 24 Multiply by 2. 5x – 2y = 11 (+)15x – 6y = 33 Multiply by 3. No Solution and Infinite Solutions Solve the Test ItemMultiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable. 19x = 57 x = 3 Example 6

  30. Replace x with 3 and solve for y. No Solution and Infinite Solutions 2x + 3y = 12 First equation 2(3) + 3y = 12 Replace x with 3. 6 + 3y = 12 Multiply. 3y = 6 Subtract 6 from each side. y = 2 Divide each side by 3. Answer: The solution is (3, 2). The correct answer is D. Example 6

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