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.