21 - GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES.pptx
- 2. LESSON OBJECTIVES: Graph systems of linear equations in two variables using slope and y-intercept and x- and y-intercepts methods. Categorize a given system of linear equations in two variables has graphs that are parallel, intersecting, and coinciding.
- 3. USING SLOPE AND Y-INTERCEPT 1. First, determine the slope and the y-intercept of the two equations. 2. Then, graph the equation using the slope and y-intercept in one rectangular coordinate plane.
- 4. EXAMPLE: Draw the graph of the following system of equations: 𝒙 + 𝒚 = 𝟑 𝟐𝒙 − 𝒚 = 𝟒 Equation Slope-intercept form (y = mx + b) Slope (m) Y-intercept (b) 𝒙 + 𝒚 = 𝟑 𝟐𝒙 − 𝒚 = 𝟒
- 5. Plot the y-intercepts and slopes. Eq. 1: x + y = 3 m = -1 b = 3 or (0, 3) Eq. 2: 2x – y = 4 m = 2 b = -4 or (0, -4) (0, 3) (1, 2) x + y = 3 (0, -4) (1, -2) 2x – y = 4
- 6. USING X- AND Y-INTERCEPTS. 1. First, determine the x-intercept and the y-intercept of the two equations. We set y = 0 to get the x-intercept and x = 0 to get the y-intercept. 2. Then, graph the equation using the x- intercept and y-intercept in one rectangular coordinate plane.
- 7. Draw the graph of the following system of equations: 2x + y = -4 and 2x + y = 2 Eq. x-intercept y-intercept Eq. 1 Eq. 2
- 8. Plot the x- and y-intercepts. Eq. 1: 2x + y = -4 x-int. y-int. (-2, 0) (0, -4) Eq. 2: 2x + y = 2 x-int. y-int. (1, 0) (0, 2) (-2, 0) (0, 2) 2x + y = 2 (0, -4) (1, 0) 2x + y = -4
- 9. A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES HAS: 1. One solution only if their graphs intersects at exactly one point. 2. Infinitely many solutions if their graphs coincide. 3. No solutions if their graphs do not intersect at all.
- 11. CONSISTENT AND INDEPENDENT SYSTEM OF EQUATIONS This type of system of equations has exactly one solution. Slopes are defined but not equal. Y-intercepts could be or could not be equal. The graphs of the two equations intersect at a point.
- 12. EXAMPLE: Draw the graph of the following system of equations: 𝟐𝒙 − 𝒚 = −𝟐 𝒙 + 𝒚 = −𝟒 Equation Slope-intercept form (y = mx + b) Slope (m) Y-intercept (b) 𝟐𝒙 − 𝒚 = −𝟐 𝒚 = 𝟐𝒙 + 𝟐 𝒙 + 𝒚 = −4 𝒚 = −𝒙 − 𝟒
- 13. Plot the y-intercepts and slopes. Eq. 1: 𝟐𝒙 − 𝒚 = −𝟐 m = 2 b = 2 or (0, 2) Eq. 2: 𝒙 + 𝒚 = −𝟒 m = −1 b = -4 or (0, -4) (0, 2) (1, 4) 2x – y = -2 (1, -5) (0, -4) x + y = -4 SOLUTION
- 14. CONSISTENT AND DEPENDENT SYSTEM OF EQUATIONS This type of system of equations has infinitely many solutions. Slopes are defined and equal. Y-intercepts are also equal. The graphs of the two equations coincide.
- 15. EXAMPLE: Draw the graph of the following system of equations: 𝒙 + 𝟑𝒚 = 𝟔 𝟐𝒙 + 𝟔𝒚 = 𝟏𝟐 Equation Slope-intercept form (y = mx + b) Slope (m) Y-intercept (b) 𝒙 + 𝟑𝒚 = 𝟔 𝒚 = − 𝟏 𝟑 𝒙 + 𝟐 𝟐𝒙 + 𝟔𝒚 = 𝟏𝟐 𝒚 = − 𝟏 𝟑 𝒙 + 𝟐
- 16. Plot the y-intercepts and slopes. Eq. 1: 𝒙 + 𝟑𝒚 = 𝟔 m = − 1 3 b = 2 or (0, 2) Eq. 2: 𝟐𝒙 + 𝟔𝒚 = 𝟏𝟐 m = − 1 3 b = 2 or (0, 2) (0, 2) (3, 1) x + 3y = 6 2x + 6y = 12
- 17. INCONSISTENT SYSTEM OF EQUATIONS This type of system of equations has no solutions. Slopes are defined and equal. Y-intercepts are not equal. The graphs of the two equations are parallel.
- 18. EXAMPLE: Draw the graph of the following system of equations: 𝟐𝒙 − 𝟑𝒚 = 𝟔 𝟐𝒙 − 𝟑𝒚 = −𝟗 Equation Slope-intercept form (y = mx + b) Slope (m) Y-intercept (b) 𝟐𝒙 − 𝟑𝒚 = 𝟔 𝒚 = 𝟐 𝟑 𝒙 − 𝟐 𝟐𝒙 − 𝟑𝒚 = −𝟗 𝒚 = 𝟐 𝟑 𝒙 + 𝟑
- 19. Plot the y-intercepts and slopes. Eq. 1: 𝟐𝒙 − 𝟑𝒚 = 𝟔 m = 2 3 b = -2 or (0, -2) Eq. 2: 𝟐𝒙 − 𝟑𝒚 = −𝟗 m = 2 3 b = 3 or (0, 3) (0, -2) (3, 0) 2x – 3y = 6 (3, 5) (0, 3) 2x – 3y = -9
- 20. WHAT’S NEXT??? Solve system of linear equations algebraically by substitution method.