Equations of Lines: Problems with Solutions

Find equations and slope of lines; questions and problems with solutions are presented.

Problem 1:

Find the equation of the line that passes through the points (-1 , 0) and (-4 , 12).

Problem 2:

What is the equation of the line through the points (-2 , 0) and (-2 , 4).

Problem 3:

Find the equation of the line that passes through the points (7 , 5) and (-9 , 5).

Problem 4:

Find the equation of the line through the point (3 , 4) and parallel to the x axis.

Problem 5:

What is the equation of the line through the point (-3 , 2) and has x intercept at x = -1.

Problem 6:

Find the equation of the line that has an x intercept at x = - 4 and y intercept at y = 5.

Problem 7:

What is the equation of the line through the point (-1 , 0) and perpendicular to the line y = 9.

Problem 8:

Find the slope, the x and y intercepts of the line given by the equation: -3 x + 5 y = 8.

Problem 9:

Find the slope intercept form for the line given by its equation: x / 4 - y / 5 = 3.

Problem 10:

Are the lines x = -3 and x = 0 parallel or perpendicular?

Problem 11:

For what values of b the point (2 , 2 b) is on the line with equation x - 4 y = 6

Solutions to the Above Problems

Solution to Problem 1:

The slope of the line is given by \[ m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{12 - 0}{-4 - (-1)} = \dfrac{-12}{3} = -4 \] We now write the equation of the line in point-slope form: \[ y - y_1 = m (x - x_1) \] \[ y - 0 = -4(x - (-1)) \] Simplify and write the equation in general form: \[ y + 4x = -4 \]

Solution to Problem 2:

The two points have the same x coordinate and are on the same vertical line whose equation is \[ x = - 2 \]

Solution to Problem 3:

The two points have the same y coordinate and are on the same horizontal line whose equation is \[ y = 5 \]

Solution to Problem 4:

A line parallel to the axis has equation of the form y = constant. Since the line we are trying to find passes through (3 , 4), then the equation of the line is given by: \[ y = 4 \]

Solution to Problem 5:

The \(x\)-intercept is the point \((-1, 0)\). The slope of the line is given by: \[ m = \dfrac{2 - 0}{-3 - (-1)} = \dfrac{2}{-2} = -1 \] The point-slope form of the line is \[ y - 0 = -1 (x - (-1)) \] The equation can be written as \[ y = -x - 1 \]

Solution to Problem 6:

The \(x\) and \(y\) intercepts are the points \((-4, 0)\) and \((0, 5)\). The slope of the line is given by: \[ m = \dfrac{5 - 0}{0 - (-4)} = \dfrac{5}{4} \] The point-slope form of the line is \[ y - 5 = \dfrac{5}{4}(x - 0) \] Multiply all terms by 4 and simplify: \[ 4y - 20 = 5x \]

Solution to Problem 7:

The line \( y = 9 \) is a horizontal line (parallel to the x axis). The line that is perpendicular to the line \( y = 9 \) have the form \( x = constant \). Since the (-1 , 0) is a point on this line, the equation is given by \[ x = -1 \]

Solution to Problem 8:

To find the slope of the given line, we first write it in slope-intercept form. \[ 5y = 3x + 8 \] \[ y = \dfrac{3}{5}x + \dfrac{8}{5} \] The slope is equal to \(\dfrac{3}{5}\). The \(y\)-intercept is found by setting \(x = 0\) in the equation and solving for \(y\). Hence, the \(y\)-intercept is at \(y = \dfrac{8}{5}\). The \(x\)-intercept is found by setting \(y = 0\) and solving for \(x\). Hence, the \(x\)-intercept is at \(x = -\dfrac{8}{3}\).

Solution to Problem 9:

Given the equation \[ \dfrac{x}{4} - \dfrac{y}{5} = 3 \] Keep only the term in \(y\) on the left side of the equation: \[ - \dfrac{y}{5} = 3 - \dfrac{x}{4} \] Multiply all terms by \(-5\): \[ y = \dfrac{5}{4} x - 15 \]

Solution to Problem 10:

The line \( x = - 3 \) is parallel to the y axis and the line \( x = 0 \) is the y axis. The two lines are parallel.

Solution to Problem 11:

For a point to be on a line, its coordinates must satisfy the equation of the line. \[ 2 - 4(2b) = 6 \] Solve for \(b\): \[ b = -\dfrac{1}{2} \]