Linear Inequalities in Two Variables Graphing Inequalities The
Linear Inequalities in Two Variables
Graphing Inequalities • The solution set for an inequality contain many ordered pairs. The graphs of these ordered pairs fill in an area of the coordinate plane and may or may not include points on the boundary line. BACK
Graph x < -3 Sketch x = -3 Now pick a test point on one side of the dotted line like (0, 0) BACK
Test a Point • Take the point (0, 0) and plug in the x value in x < -3 • x < -3 • 0 < -3 Since it’s false, shade the side opposite of (0, 0).
Graph x < -3 x = -3 Shade the area with true solutions!
Graph y ≤ 4 Sketch y = 4 y=4 Now pick a point on one side of the solid line (0, 0)
Test a Point • Take the point (0, 0) and plug in the y value in y ≤ 4 • y ≤ 4 • 0 ≤ 4 Since it’s True, shade the side that (0, 0) is on.
Graph x < -3 & y ≤ 4 y=4 BACK
Graph x < -3 & y ≤ 4 y≤ 4 x < -3 What’s the difference between the dotted line and the solid line?
Dotted or Solid Lines? ? ? • Use a solid line if your equation contains any part of an equal sign ( =, ≤, ≥ ) to show that points that fall on the line are include in the solution. • Use a dotted or dashed line if you have < or > to show that the points on this line are not part of the solution area. BACK
Graph x + y < 3 Sketch y = -x + 3 y= -x +3 Now pick a point on one side of the dotted line (0, 0) BACK
Test a Point • Take the point (0, 0) and plug in the values in y < -x + 3 • y < -x + 3 • 0 < -0 + 3 • 0 < 3 Since it’s True, shade the side that (0, 0) is on.
Graph x + y < 3 y= -x +3 BACK
You try this one • Graph y ≤ 2 x - 1 You try this one BACK
Graph y ≤ 2 x - 1 Sketch y = 2 x - 1 y= 2 x - 1 Now pick a test point on one side of the dotted line (-1, 0)
Test a Point • Take the point (-1, 0) and plug in the values in y ≤ 2 x - 1 • y ≤ 2 x - 1 • 0 ≤ 2(-1) -1 • 0 ≤ -3 Since it’s False, shade the opposite side of (-1, 0). on.
Graph y > 2 x - 1 y= 2 x - 1 BACK
Graph the following linear system of inequalities. Use the slope and yintercept to plot the two lines. y Draw in the line. For < use a dashed line. x Pick a point and test it in the inequality. Shade the appropriate region. BACK
Graph the following linear system of inequalities. y x The region below the line should be shaded. BACK
Graph the following linear system of inequalities. y x The solution to this system of inequalities is the region where the solutions to each inequality overlap. This is the region above or to the left of the green line and below or to the left of the blue line. Shade in that region. BACK
Systems of Equations Solving by Graphing
Equivalent Slopes Sometimes a point will not fit on your graph and may need changed to an equivalent number.
Systems of Equations • One way to solve equations that involve two different variables is by graphing the lines of both equations on a coordinate plane. • If the two lines cross the solution for both variables is the coordinate of the point where they intersect.
y = 2 x + 0 & y = -1 x + 3 Slope = 1/-1 y-intercept= 0 Slope = 2/1 Up 2 Up 1 (1, 2) and The solution is the point they cross at (1, 2) right 1 left 1 y-intercept= +3
y = x - 3 & y = -3 x + 1 Slope = 3/-1 y-intercept= -3 Slope = 1/1 y-intercept= +1 The solution is the point they cross at (1, -2)
y =-2 x + 4 & y = 2 x + 0 Slope = 2/1 y-intercept= 4 Slope = 2/-1 y-intercept= 0 The solution is the point they cross at (1, 2)
Graph y = x -3 y=x+2 Solution= none
Two lines in a plane will: • be parallel • coincide • or intersect.
IDENTIFYING THE NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y y y x x x Lines intersect Lines are parallel Lines coincide one solution no solution infinitely many solutions
Identifying the solutions • 1. ) They Intersect One Solution • 2. ) They are Parallel None/ No Solution • 3. ) They Coincide Infinite Solutions
Solve by Graphing: x + 2 y = 7 x+y=1
y = -1/2 x + 7/2 y = -x + 1 7/2 = 3 1/2 (-5, 6) Notice the intersection is at a single point.
Name the Solution
Name the Solution
Name the Solution
y=x+5 y = -x -1 2 = -3 + 5 2 = -(-3) – 1 2=2 2=3– 1 2=2 This screen shows two lines which have exactly one point in common. The common point when substituted into the equation of each line makes that equation true. The common point is (-3, 2). Try it and see.
y=x+5 y = -x -1 2 = -3 + 5 2 = -(-3) – 1 2=2 2=3– 1 2=2 This system of equations is called consistent because it has at least one ordered pair that satisfies both equations. A system of equations that has exactly one solution is called independent.
This graph crosses the y-axis at 2 and has a slope of – 2 (down two and right one). Thus the equation of this line is y = -2 x + 2. What do you notice about the graph on the right? It appears to be the same as the graph on the left and would also have the equation y = -2 x + 2.
Line 1 crosses the y-axis at 3 and has a slope of 3. Therefore, the equation of line 1 is y = 3 x + 3. Line 2 crosses the y-axis at 6 and has a slope of 3. Therefore, the equation of line 2 is y = 3 x + 6. Parallel lines have the same slope, in this case 3. However, parallel lines have different y-intercepts. In our example, one yintercept is at 3 and the other y-intercept is at 6. Parallel lines never intersect. Therefore parallel lines have no points in common and are called inconsistent.
1) Write the equations of the lines in slope-intercept form. 2) Use the slope and y-intercept of each line to plot two points for each line on the same graph. 3) Draw in each line on the graph. 4) Determine the point of intersection and write this point as an ordered pair.
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.
y The two equations in slopeintercept form are: x Plot points for each line. Draw in the lines. These two equations represent the same line. Therefore, this system of equations has infinitely many solutions.
y The two equations in slopeintercept form are: x Plot points for each line. Draw in the lines. This system of equations represents two parallel lines. This system of equations has no solution two lines have no points in common. because these
y The two equations in slopeintercept form are: x Plot points for each line. Draw in the lines. This system of equations represents two intersecting lines. The solution to this system of equations is a single point (3, 0).
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