Algebra: Root of a Linear Equation In One Unknown (Fractional)

Analytic Geometry: Graphs of Intersecting And Parallel Lines

The following illustrated problems are about:

• intersecting lines (perpendicular and non-perpendicular)
• parallel lines

All the calculated values are given so as to allow flexibility in presenting the problems in different ways.

1-1 Graphs of Perpendicular Bisector of a Line Segment

Find the point at which the line x - 2y + 3 = 0 bisects perpendicularly the line segment formed by the x-intercept and y-intercept of the line 2x + y - 4 = 0.

• point of intersection (midpoint of line segment AB): (1,2)
• slope of the line x - 2y + 3 = 0: m = 1/2
• angle of inclination of line x - 2y + 3 = 0: 26.6°
• points on the line x - 2y + 3 = 0: {(-5, -1), (-4, -1/2), (-2, 1/2), (-1, 1), (0, 3/2), (-3, 0), (1, 2), (2, 5/2), (3, 3), (4, 7/2), (5, 4)}
• slope of the line 2x + y - 4 = 0: m = -2
• angle of inclination of line 2x + y - 4 = 0: 116.6°
• points on the line 2x + y - 4 = 0: {(-5, 14), (-4, 12), (-3, 10), (-2, 8), (-1, 6), (0, 4), (2, 0), (3, -2), (4, -4), (5, -6)}

Find the point at which the line x + 5y + 12 = 0 bisects perpendicularly the line segment formed by the x-intercept and y-intercept of the line 5x - y - 5 = 0.

• point of intersection (midpoint of line segment AB): (1/2, -5/2)
• slope of the line x + 5y + 12 = 0: m = -1/5
• angle of inclination of line x + 5y + 12 = 0: 168.7°
• points on the line x + 5y + 12 = 0: {(-5, -7/5), (-4, -8/5), (-3, -9/5), (-2, -2), (-1, -11/5), (0, -12/5), (-12, 0), (1, -13/5), (2, -14/5), (3, -3), (4, -16/5), (5, -17/5)}
• slope of the line 5x - y - 5 = 0: m = 5
• angle of inclination of line 5x - y - 5 = 0: 78.7°
• points on the line 5x - y - 5 = 0: {(-5, -30), (-4, -25), (-3, -20), (-2, -15), (-1, -10), (0, -5), (1, 0), (2, 5), (3, 10), (4, 15), (5, 20)}

Find the point at which the line 5x + 3y = 0 bisects perpendicularly the line segment AB.

• point of intersection (midpoint of line segment AB): (-3/2, 5/2)
• slope of the line 5x + 3y = 0: m = -5/3
• angle of inclination of line 5x + 3y = 0: 120.96°
• points on the line 5x + 3y = 0: {(-5, 25/3), (-4, 20/3), (-3, 5), (-2, 10/3), (-1, 5/3), (0, 0), (1, -5/3), (2, -10/3), (3, -5), (4, -20/3), (5, -25/3)}
• slope of the line segment AB: m = 3/5
• equation of line segment AB: 3x - 5y + 17 = 0

1-2 Graphs of Perpendicular Lines

Find the point of intersection of lines 8x + 5y + 13 = 0 and 5x - 8y - 3 = 0.

• point of intersection: (-1, -1)
• slope of the line 8x + 5y + 13 = 0: m = -8/5
• angle of inclination of line 8x + 5y + 13 = 0: 122.0°
• points on the line 8x + 5y + 13 = 0: {(-5, 27/5), (-4, 19/5), (-3, 11/5), (-2, 3/5), (0, -13/5), (-13/8, 0), (1, -21/5) (2, -29/5), (3, -37/5), (4, -9), (5, -53/5)}
• slope of the line 5x - 8y - 3 = 0: m = 5/8
• angle of inclination of line 5x - 8y - 3 = 0: 32.0°
• points on the line 5x - 8y - 3 = 0: {(-5, -7/2), (-4, -23/8), (-3, -9/4), (-2, -13/8), (0, -3/8), (3/5, 0), (1, 1/4), (2, 7/8), (3, 3/2), (4, 17/8), (5, 11/4))}

Find the point of intersection of lines x + 5y - 5 = 0 and 5x - y - 2 = 0.

• point of intersection: (15/26, 23/26)
• slope of the line x + 5y - 5 = 0: m = -1/5
• angle of inclination of line x + 5y - 5 = 0: 168.7°
• points on the line x + 5y - 5 = 0: {(-5, 2), (-4, 9/5), (-3, 8/5), (-2, 7/5), (-1, 6/5), (0, 1), (5, 0), (1, 4/5), (2, 3/5), (3, 2/5), (4, 1/5)}
• slope of the line 5x - y - 2 = 0: m = 5
• angle of inclination of line 5x - y - 2 = 0: 78.7°
• points on the line 5x - y - 2 = 0: {(-5, -27), (-4, -22), (-3, -17), (-2, -12), (-1, -7), (0, -2), (2/5, 0), (1, 3), (2, 8), (3, 13), (4, 18), (5, 23)}

1-3 Graphs of Intersecting Non-Perpendicular Lines

Find the point of intersection of lines 5x + 6y - 8 = 0 and 2x + y + 1 = 0.

• point of intersection: (-2, 3)
• slope of the line 5x + 6y - 8 = 0: m = -5/6
• angle of inclination of line 5x + 6y - 8 = 0: 140.2°
• points on the line 5x + 6y - 8 = 0: {(-5, 11/2), (-4, 14/3), (-3, 23/6), (-1, 13/6), (0, 4/3), (8/5, 0), (1, 1/2), (2, -1/3), (3, -7/6), (4, -2), (5, -17/6)}
• slope of the line 2x + y + 1 = 0: m = -2
• angle of inclination of line 2x + y + 1 = 0: 116.6°
• points on the line 2x + y + 1 = 0: {(-5, 9), (-4, 7), (-3, 5), (-1, 1), (0, -1), (-1/2, 0), (1, -3), (2, -5), (3, -7), (4, -9), (5, -11)}

Find the point of intersection of lines 5x + y + 16 = 0 and 3x - y + 8 = 0.

• point of intersection: (-3, -1)
• slope of the line 5x + y + 16 = 0: m = -5
• angle of inclination of line 5x + y + 16 = 0: 101.3°
• points on the line 5x + y + 16 = 0: {(-5, 9), (-4, 4), (-2, -6), (-1, -11), (0, -16), (-16/5, 0), (1, -21), (2, -26), (3, -31), (4, -36), (5, -41)}
• slope of the line 3x - y + 8 = 0: m = 3
• angle of inclination of line 3x - y + 8 = 0: 71.6°
• points on the line 3x - y + 8 = 0: {(-5, -7), (-4, -4), (-2, 2), (-1, 5), (0, 8), (-8/3, 0), (1, 11), (2, 14), (3, 17), (4, 20), (5, 23)}

2 Graphs of Parallel Lines

Show that lines 3x + 5y - 4 = 0 and 3x + 5y + 15 = 0 are parallel.

• points on the line 3x + 5y - 4 = 0: {(-5, 19/5), (-4, 16/5), (-3, 13/5), (-2, 2), (-1, 7/5), (0, 4/5), (4/3, 0), (1, 1/5), (2, -2/5), (3, -1), (4, -8/5), (5, -11/5)}
• slope of the lines: m = -3/5
• angle of inclination of the lines: 149.0°
• points on the line 3x + 5y + 15 = 0: {(-5, 0), (-4, -3/5), (-3, -6/5), (-2, -9/5), (-1, -12/5), (0, -3), (1, -18/5), (2, -21/5), (3, -24/5), (4, -27/5), (5, -6)}

Show that lines 4x - 3y + 1 = 0 and 4x - 3y - 3 = 0 are parallel.

• points on the line 4x - 3y + 1 = 0: {(-5, -19/3), (-4, -5), (-3, -11/3), (-2, -7/3), (-1, -1), (0, 1/3), (-1/4, 0), (1, 5/3), (2, 3), (3, 13/3), (4, 17/3), (5, 7)}
• slope of the lines: m = 4/3
• angle of inclination of the lines: 53.1°
• points on the line 4x - 3y - 3 = 0: {(-5, -23/3), (-4, -19/3), (-3, -5), (-2, -11/3), (-1, -7/3), (0, -1), (3/4, 0), (1, 1/3), (2, 5/3), (3, 3), (4, 13/3), (5, 17/3)}