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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.

Verification of Points on a Line

Given Points:

• Point A: (1, 1)
• Point B: (1, 2)
• Point C: (1, 3)
• Point D: (1, 4)

Plotting the Points:

Let's plot the given points on a coordinate plane.

• Point A (1, 1)
• Point B (1, 2)
• Point C (1, 3)
• Point D (1, 4)

These points are plotted on the y-axis where the x-coordinate is always 1.

Verification:

We observe that all the points lie on a vertical line parallel to the y-axis.

• The x-coordinate is constant (1) for all points.
• The y-coordinates are different but increase sequentially (1, 2, 3, 4).

Conclusion:

The given points A, B, C, and D lie on a vertical line parallel to the y-axis.

Line Name:

The line passing through these points is a vertical line.

This concludes the verification process for the given points. If you have any further questions or need clarification, feel free to ask.

To plot the given points K(1,3)K(1,3), L(2,3)L(2,3), M(3,3)M(3,3), and N(4,3)N(4,3), we can plot them on a Cartesian coordinate system:

• Point K(1,3)K(1,3)
• Point L(2,3)L(2,3)
• Point M(3,3)M(3,3)
• Point N(4,3)N(4,3)

All these points lie on the line y=3y=3, which is a horizontal line passing through the y-coordinate 3. This line is commonly known as the horizontal line y=3y=3.

So, the points KK, LL, MM, and NN all lie on the horizontal line y=3y=3.