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Solving Systems of Equations - Methods and Examples

Learn how to solve systems of linear equations using substitution and elimination methods with helpful examples and practice questions.

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Solving Systems of Equations - Methods and Examples

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 2) CCSS Then/Now New Vocabulary Example 1: Solve by Using a Table Example 2: Solve by Graphing Example 3: Classify Systems Concept Summary: Characteristics of Linear Systems Key Concept: Substitution Method Example 4: Real-World Example: Use the Substitution Method Key Concept: Elimination Method Example 5: Solve by Using Elimination Example 6: Standardized Test Example: No Solution and Infinite Solutions Concept Summary: Solving Systems of Equations Lesson Menu

  3. Find the domain and range of the relation {(–4, 1), (0, 0), (1, –4), (2, 0), (–2, 0)}. Determine whether the relation is a function. A. D= {–4, –2, 0, 1, 2}, R= {–4, 0,1}; yes B. D= {0, 1, 2}, R= {0, 1}; yes C. D= {–4, 0, 1}, R= {–4, –2, 0, 1, 2}; no D. D= {–2, –4}; R= {–4, 0, 1}; yes 5-Minute Check 1

  4. Find the value of f(4) for f(x) = 8 – x – x2. A. 28 B. 12 C. –12 D. –16 5-Minute Check 2

  5. A. B. C.2 D.7 Find the slope of the line that passes through (5, 7) and (–1, 0). 5-Minute Check 3

  6. Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6. A.y = –3x + 6 B.y = –3x – 6 C.y = 3x + 6 D.y = 2x + 6 5-Minute Check 4

  7. Identify the type of function represented by the equation y = 4x2 + 6. A. absolute value B. linear C. piecewise-defined D. quadratic 5-Minute Check 6

  8. Mathematical Practices 2 Reason abstractly and quantitatively. 6 Attend to precision. CCSS

  9. You graphed and solved linear equations. • Solve systems of linear equations graphically. • Solve systems of linear equations algebraically. Then/Now

  10. break-even point • system of equations • consistent • inconsistent • independent • dependent • substitution method • elimination method Vocabulary

  11. Solve the system of equations by completing a table. x + y = 3–2x + y = –6 Solve by Using a Table Solve for y in each equation. x + y = 3 y = –x + 3 –2x + y = –6 y = 2x – 6 Example 1

  12. Solve by Using a Table Use a table to find the solution that satisfies both equations. Answer: The solution to the system is (3, 0). Example 1

  13. What is the solution of the system of equations? x + y = 2x – 3y = –6 A. (1, 1) B. (0, 2) C. (2, 0) D. (–4, 6) Example 1

  14. 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

  15. 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

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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Concept

  26. 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

  27. 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

  28. 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

  29. D. Graph the system of equations below. Which statement is not true? f(x) = x + 2 g(x) = x + 4 A. f(x) and g(x) areconsistent and dependent. B.f(x) and g(x) areinconsistent. C. f(x) and h(x) areconsistent and independent. D.g(x) and h(x) areconsistent. Example 3

  30. Concept

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. Concept

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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

  43. 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

  44. A. B.(1, 2) C.(–5, 4) D.no solution Solve the system of equations.x + 3y = 72x + 5y = 10 Example 6

  45. Concept

  46. End of the Lesson

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