To determine if the points (5, 4) and (4, 5) represent the same point, we can compare their coordinates.

The point (5, 4) has coordinates (x, y) = (5, 4), while the point (4, 5) has coordinates (x, y) = (4, 5).

Since the order of the coordinates matters, these points are different. The first point is located at (5, 4), and the second point is located at (4, 5). Therefore, they do not represent the same point in a Cartesian coordinate system.

Solution: Finding Coordinates of Intercepts for a Line

• Given Points:

  • Point A (2, 1)
  • Point B (1, 2)

• Finding Slope (m):

  • Slope (m) = (Change in y) / (Change in x)
  • m=(2−1)(1−2)=−1m=(1−2)(2−1)=−1

• Using Point-Slope Form:

  • Equation of the line: y−y1=m(x−x1)y−y1=m(x−x1) where (x1,y1)(x1,y1) is a point on the line.
  • Substituting values of Point A (2, 1): y−1=−1(x−2)y−1=−1(x−2)

• Solving for y-intercept (x = 0):

  • Let x = 0: y−1=−1(0−2)y−1=−1(0−2)
  • y−1=2y−1=2
  • y=3y=3
  • Therefore, y-intercept is (0, 3)

• Solving for x-intercept (y = 0):

  • Let y = 0: 0−1=−1(x−2)0−1=−1(x−2)
  • −1=−x+2−1=−x+2
  • x=3x=3
  • Therefore, x-intercept is (3, 0)

• Summary:

  • y-intercept: (0, 3)
  • x-intercept: (3, 0)

This concludes the solution for finding the coordinates of the points where the line intersects the x-axis and y-axis.

Line Passing through (2,3) and (3,2) - Coordinates of Intercepts

1. Finding the Slope (m) of the Line:

• Formula for Slope (m): m=y2−y1x2−x1m=x2−x1y2−y1

• Given Points: P1(x1,y1)=(2,3)P1(x1,y1)=(2,3) P2(x2,y2)=(3,2)P2(x2,y2)=(3,2)

• Calculation: m=2−33−2=−1m=3−22−3=−1

2. Writing the Equation of the Line:

• Point-Slope Form: y−y1=m⋅(x−x1)y−y1=m⋅(x−x1)

• Substitute Values: y−3=−1⋅(x−2)y−3=−1⋅(x−2)

• Simplify: y=−x+1y=−x+1

3. Finding the x-Intercept:

• Definition: The x-intercept is where the line crosses the x-axis (y=0y=0).

• Substitute y=0y=0 in the Equation: 0=−x+10=−x+1

• Solve for x: x=1x=1

• Coordinates of x-Intercept: (1,0)(1,0)

4. Finding the y-Intercept:

• Definition: The y-intercept is where the line crosses the y-axis (x=0x=0).

• Substitute x=0x=0 in the Equation: y=−0+1y=−0+1

• Coordinates of y-Intercept: (0,1)(0,1)

5. Summary:

• Equation of the Line: y=−x+1y=−x+1

• x-Intercept: (1,0)(1,0)

• y-Intercept: (0,1)(0,1)

This completes the analysis of the line passing through the points (2,3) and (3,2) with the determination of the x and y intercepts.

Line Passing through (2,3) and (3,2) - Coordinates of Intercepts

1. Finding the Slope (m) of the Line:

• Formula for Slope (m): m=y2−y1x2−x1m=x2−x1y2−y1

• Given Points: P1(x1,y1)=(2,3)P1(x1,y1)=(2,3) P2(x2,y2)=(3,2)P2(x2,y2)=(3,2)

• Calculation: m=2−33−2=−1m=3−22−3=−1

2. Writing the Equation of the Line:

• Point-Slope Form: y−y1=m⋅(x−x1)y−y1=m⋅(x−x1)

• Substitute Values: y−3=−1⋅(x−2)y−3=−1⋅(x−2)

• Simplify: y=−x+1y=−x+1

3. Finding the x-Intercept:

• Definition: The x-intercept is where the line crosses the x-axis (y=0y=0).

• Substitute y=0y=0 in the Equation: 0=−x+10=−x+1

• Solve for x: x=1x=1

• Coordinates of x-Intercept: (1,0)(1,0)

4. Finding the y-Intercept:

• Definition: The y-intercept is where the line crosses the y-axis (x=0x=0).

• Substitute x=0x=0 in the Equation: y=−0+1y=−0+1

• Coordinates of y-Intercept: (0,1)(0,1)

5. Summary:

• Equation of the Line: y=−x+1y=−x+1

• x-Intercept: (1,0)(1,0)

• y-Intercept: (0,1)(0,1)

This completes the analysis of the line passing through the points (2,3) and (3,2) with the determination of the x and y intercepts.