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Solving Systems of Equations by Graphing

Solving Systems of Equations by Graphing. One option to solving a system of two linear equations in two variables is to solve it graphically. To solve a system graphically, graph both equations on the same coordinate system. Their intersection is the solution to the system of equations.

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Solving Systems of Equations by Graphing

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  1. Solving Systems of Equations by Graphing One option to solving a system of two linear equations in two variables is to solve it graphically. To solve a system graphically, graph both equations on the same coordinate system. Their intersection is the solution to the system of equations.

  2. Example: Given the system below, solve for x and y to find the point of intersection. 8x - 2y  = 10 3x + y =2

  3. Solution: Start by writing both equations in slope-intercept form, y = mx + b. 8x - 2y  = 10 -2y = -8x + 10 y = 4x- 5 And 3x + y = 2 y = -3x + 2

  4. Graph the lines on the same coordinate plane The two lines intersect at (1, -1), so the solution to the system is the point (1, -1).

  5. Practice Problem #1 The solution to this system is in which quadrant? y = -2x- 4 y = x A. I B. II C. III D. IV Check 

  6. Practice Problem #2 2. Leslie joins a fitness club that has a membership fee of $20 plus $15 per month. Ray’s club has a fee of $40 and charges $10 per month. • In how many months will the two clubs cost the same? • What is that amount? • Hint: You’ll need to adjust your viewing window; • [ZOOM] 0:ZoomFit Check 

  7. Practice Problem #3 3. The graph of a system of linear equations is shown below. What is the solution of the system? • Write your answer as an ordered pair. Check 

  8. Practice Problem #4 Solve this system of equations. –x + y = 1 x + y = 3 Check 

  9. Practice Problem #5 The box office took in a total of $2905 in paid admissions for the high−school musical. Adult tickets cost $8 each, and student tickets cost $3 each. If 560 people attended the show, how many were students? Let s = the number of students attending, and let a = the number of adults attending. Which system of equations can be used to solve this problem? A. 3a + 8s = 2905 C. 3a + 8s = 560 a = 560 – s a + s = 2905 B. 8a + 3s = 2905 D. 8a + 3s = 560 a + s = 560 a + s = 2905 Check 

  10. Practice Problem #6 Carlos invested a total of $6500.00 for a year and earned a simple interest of $305. He invested part at 4% and the rest at 5%. How much did he invest at each rate? Let x = the amount invested at 4%, and let y = the amount invested at 5%. Which two equations can be used to solve this problem? A. x + y = 6500 C. .05x + .04y = 6500 .04x + .05y = 305 x + y = 305 B. .04x + .05y = 6500 D. .05x + .04y = 305 x + y = 305 x + y = 6500 Check 

  11. Practice Problem #7 Katrina has 10 nickels and dimes that total $0.80 in her purse. Which system of equations could be used to determine the number of nickels, n, and the number of dimes, d, Katrina has in her purse? A. n + d = 0.80 C. n + d = 0.80 0.05n + 0.10d = 10 0.5n + 0.10d = 10 B. n + d = 10 D. n + d = 10 0.05n + 0.10d = 0.80  0.10n + .05d = 0.80 Check 

  12. Practice Problem #8 Mr. Harrington's farm is 65 acres larger then Mr. Hargrove's farm. Together the farms contain 1288 acres. Which system of equations could be used to determine the number of acres in Mr. Hargrove's farm, x, and the number of acres in Mr. Harrington's farm, y? A. y = x + 65 C. x + y = 65 x + y = 1288 x + y = 1288 B. x = y + 65 D. y = x + 1288 x + y = 1288 x = y + 65 Check 

  13. Practice Problem #9 The cost of renting a car for 1 day at Cars Plus is $20 plus 10 cents per mile driven. The cost of renting a car for 1 day at Need-A-Car is $20 plus 15 cents per mile driven. In a graph of the cost of a car rental, what does the cost per mile driven represent? A. The x-intercept B. The y-intercept C. The slope D. The origin Check 

  14. Practice Problem #10 The perimeter of a rectangular volleyball court is 180 feet. The court’s width, w, is half its length, l. Which system of linear equations could be used to determine the dimensions, in feet, of the volleyball court? A. l + w = 180 w = ½ l B 2l + 2w = 180 w = ½ l C 2l + 2w = 180 l = ½ w End Show

  15. Answer to #1 C – The lines intersect in Quadrant III  Back

  16. Answer to #2 • A) 4 months • B) $80 • The intersection (4, 80) means that, in 4 months, both clubs will cost a total of $80.  Back

  17. Answer to #3 The solution is the point of intersection, which is (2, 2).  Back

  18. Answer to #4 Rewrite each equation in slope-intercept form & enter them in your Y= screen. -x + y = 1 y = x + 1 x + y = 3 y = -x + 3 Graph and find the point of intersection using [2nd] [TRACE] 5 [ENTER] [ENTER] [ENTER] The solution is (1, 2).  Back

  19. Answer to #5 • The correct answer is B.  Back

  20. Answer to #6 The correct answer is A.  Back

  21. Answer to #7 The correct answer is B.  Back

  22. Answer to #8 The correct answer is A.  Back

  23. Answer to #9 C. The cost per mile represents the slope. Slope is the ratio of the change in y compared to the change in x. Any phrase like “cost PER mile”, “cost PER ounce”, “miles PER gallon”, “(blank) PER (blank)” represents the slope.  Back

  24. Answer to #10 The correct answer is A.  Back

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