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U2 Day 1 Systems of Equations

Understand and solve systems of equations using graphing, substitution, and elimination methods to find solutions. Learn about infinite, one, or no solutions, and graph equations without a calculator.

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U2 Day 1 Systems of Equations

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  1. U2 Day 1 Systems of Equations

  2. Math3H – U2 Day 1 Systems of Equations

  3. Unit 2 Day 1 - OBJECTIVES • To understand what a system of equations is. • Be able to solve a system of equations using graphing, substitution, and elimination. • Determine whether the system has one, none, or infinite solutions. • Be able to graph equations without using a graphing calculator.

  4. x + y = 2 2x + y = 5 2y = x + 2 y = 5x - 7 6x - y = 5 Defining a System of Equations • A grouping of 2 or more equations, containing one or more variables.

  5. Point of intersection How do we “solve” a system of equations??? • By finding the point where two or more equations, intersect. x + y = 6 y = 2x 6 4 2 1 2

  6. How do we “solve” a system of equations??? • By finding the point where two or more equations, intersect. x + y = 6 y = 2x 6 (2,4) 4 2 1 2

  7. ax + by = c Standard Form of a Linear Equation NOTE: The equation ax + by = c is just another form of a linear equation. TO GRAPH: Change it to y = mx + b OR use the x and y-intercepts 

  8. WE WANT y = mx + b 3 3 3 2 y = 2 - x 3 2 y = - + 2 x 3 ax + by = c to y = mx + b 2x + 3y = 6 ax + by = c (Standard Form) -2x -2x 3y = 6 - 2x y = mx + b (Slope- Intercept)

  9. Solutions of Systems No Solution: • when lines of a graph are parallel • since they do not intersect, there is no solution

  10. Solutions of Systems Infinite Solutions: • a pair of equations that have the same slope and y-intercept.

  11. Solutions of Systems One Solution: • the lines of two equations intersect

  12. 2y + x = 8 x - 5y = 10 y = -6x + 8 y = 2x + 4 -5y = -x +6 y + 6x = 8 Examples… How many solutions? Look at the slope and y-intercepts in your group 1) 2) 3) ANS: One Solution ANS: Infinite Solutions ANS: No Solution

  13. Graphing Manually Using the y-intercept and the linear slope to graph the equation: y = 2x + 4

  14. Graphing Manually Using the y-intercept and the linear slope to graph the equation: y = 2x + 4 1. Plot the y-intercept

  15. Graphing Manually Using the y-intercept and the linear slope to graph the equation: y = 2x + 4 1. Plot the y-intercept 2. Use the slope to plot second point (rise and run)

  16. Graphing Manually Using the y-intercept and the linear slope to graph the equation: y = 2x + 4 1. Plot the y-intercept 2. Use the slope to plot second point (rise over run) 3. Draw a line connecting the two points.

  17. Graphing Manually Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y! 2x + 3y = 4

  18. 3 3 Graphing Manually Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y! 2x + 3y = 4 1. Substitute 0 for x, then solve for y x = 0 2(0) + 3y = 4 3y = 4 y = 1.33 2. Plot the point: (0, 1.33)

  19. Graphing Manually Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y! 2x + 3y = 4 3. Substitute 0 for y,then solve for x y = 0 2x + 3(0) = 4 2x = 4 x = 2 4. Plot the point: (2, 0)

  20. Graphing Manually cont. Plot two points by finding the x and y-intercepts …Substitute 0 for x and then y! 5. Draw line connecting both points.

  21. More Examples HOW MANY SOLUTIONS?? If “one solution” graph it and give the point of intersection. NON-Calculator! 1) 2) 3) 2 x + 2y = 6 y = x - 1 3 x + 2y = 8 y = 3 ANS: One Solution (6,3) ANS: No Solution ANS: Infinite Solutions

  22. SO… What do we know?  • You can graph systems of equations from standard form and slope-intercept form. • You can determine how many solutions there are: none/one/infinite • Sometimes you can tell how many solutions without graphing if you look at the slopes and intercepts!

  23. HOMEWORK  • U2 Day 1 HW

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