Introduction to Algebra

Part I

Equation of a line parallel to the given line and passing through the given point.y = 3x + 3; (1, 1)

Step 1) Since the lines are parallel, the slopes are equal. Therefore, substitute the slope from the original line into the equation of a line.

y = mx + b

y = 3x + b

Step 2) Substitute the values of x and y in the equation and find 'b' (the y-intercept).

y = 3x + b

1 = 3(1) + b

1 = 3 + b

b = 1 - 3

b = -2

Step 3) Substitute the value of 'b' into the slop-intercept form of the equation

y = 3x + (-2)

y = 3x - 2

The line looks just like the original line since they are both parallel to each other.

Equation of a line perpendicular to the given line but passing through the given point.

y = 3x + 3; (1, 1)

Step 1) Since the slope of the line is 3, the slope of any line perpendicular to this line is -1/3.

y = mx + b

y = -1/3x +b

Step 2) Substitute the values of x and y in the equation and find 'b' (the y-intercept).

y = -1/3x +b

1 = -1/3(1) +b

1 = -1/3 + b

b = 1+1/3

b = 4/3

Step 3) Substitute the value of 'b' into the slop-intercept form of the equation

y = -1/3x + 4/3

y = -1/3x + b

The line passes through the given line at an angle at 90 degrees.

What does it mean for one line to be parallel to another?

For one line to be parallel to each other, the slope of both the lines needs to be the same. A property of parallel lines is that they do not intersect each other even if they are extended to infinity. The origin of parallel lines can only be the same if one line is superimposed ...