Problem 37 - College Algebra

Question

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $$ \begin{array}{l} 4 x-7 y=10 \\ 7 x+4 y=1 \end{array} $$

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Rewrite the First Equation in Slope-Intercept Form

The equation is given as \(4x - 7y = 10\). To find the slope-intercept form \(y = mx + b\), solve for \(y\). Start by moving \(4x\) to the right side: \[-7y = -4x + 10\] Next, divide everything by \(-7\):\[y = \frac{4}{7}x - \frac{10}{7}\]The slope of the first line is \(m_1 = \frac{4}{7}\).

Step 2: Rewrite the Second Equation in Slope-Intercept Form

The equation is given as \(7x + 4y = 1\). Similar to the first equation, rearrange to get \(y\) alone:Move \(7x\) to the right side: \[4y = -7x + 1\]Divide everything by \(4\): \[y = -\frac{7}{4}x + \frac{1}{4}\]The slope of the second line is \(m_2 = -\frac{7}{4}\).

Step 3: Determine if Lines are Parallel, Perpendicular, or Neither

Two lines are parallel if their slopes are equal, \(m_1 = m_2\), and perpendicular if the product of their slopes is \(-1\), i.e., \(m_1 \times m_2 = -1\). If neither condition holds, the lines are neither.The slopes from Steps 1 and 2 are \(m_1 = \frac{4}{7}\) and \(m_2 = -\frac{7}{4}\).Check for perpendicular: Compute \(m_1 \times m_2 = \frac{4}{7} \times -\frac{7}{4} = -1\).Since the product is \(-1\), the lines are perpendicular.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a very convenient way to represent a line. It is given by the formula \(y = mx + b\). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept. The y-intercept is where the line crosses the y-axis.
To convert any linear equation into this form, you'll want to solve for \(y\). For example, consider the equation \(4x - 7y = 10\). By rearranging it, we transform it into \(y = \frac{4}{7}x - \frac{10}{7}\), showing the line's slope (\(\frac{4}{7}\)) and y-intercept.
This form makes it easy to understand how changes in \(x\) affect \(y\). If \(m\) is positive, \(y\) increases with \(x\). If \(m\) is negative, \(y\) decreases as \(x\) increases.

• Slope \(m\) indicates how steep a line is.
• The y-intercept \(b\) is where the line crosses the y-axis (\(x = 0\)).

Parallel Lines

Parallel lines are fascinating because they never meet, extending endlessly with the same slope. If two lines have the same slope, they are parallel. Their equations in slope-intercept form will have identical \(m\) values, but their \(b\) values can differ. This is because having different y-intercepts simply means they start at different points on the vertical axis.
For example, suppose you have lines with equations \(y = 2x + 3\) and \(y = 2x - 5\). The slopes \(m = 2\) for both lines show that they are parallel, even though they cross the y-axis at different points.

• Make sure the lines have equal slopes.
• The y-intercepts being different simply means they are offset along the y-axis.
• Remember: Equal slopes, parallel lines.

Perpendicular Lines

Perpendicular lines intersect at a right angle, forming a 90-degree corner. To determine if lines are perpendicular, check if the product of their slopes is \(-1\). In mathematical terms, if one line has slope \(m_1\) and the other \(m_2\), then \(m_1 \times m_2 = -1\) means the lines are perpendicular.
Take the example from our initial problem where we found slopes \(m_1 = \frac{4}{7}\) for the first line and \(m_2 = -\frac{7}{4}\) for the second line. Multiplying these slopes \(\left(\frac{4}{7}\right) \times \left(-\frac{7}{4}\right)\) gives us \(-1\), proving they are perpendicular.

• Check for the negative reciprocal relationship between the slopes.
• The product of the slopes must always be \(-1\).
• When perpendicular, lines form 90-degree angles at the intersection.