Problem 35 - Introductory Algebra for College Students

Question

In Exercises \(27-30\), determine whether the lines through each pair of points are perpendicular. $$(-5,-1) \text { and }(0,2) ;(-6,9) \text { and }(3,-6)$$

Step-by-Step Solution

Answer

Yes, the lines through the given pairs of points are perpendicular.

Step 1: Calculate Slope of Line 1

Recall that the equation to calculate the slope (m) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = (y_2 - y_1) / (x_2 - x_1)\). By substituting the given first pair of points (-5,-1) and (0,2) in the formula, we can derive the slope for line 1. It gives \(m_1 = (2 - (-1)) / (0 - (-5)) = 3/5\).

Step 2: Calculate Slope of Line 2

Using the same formula, we calculate the slope for the second pair of points (-6,9) and (3,-6). It gives \(m_2 = (-6 - 9) / (3 - (-6)) = -15/9 = -5/3\).

Step 3: Determine If Lines Are Perpendicular

Two lines are perpendicular if the product of their slopes is -1. Calculate the product of \(m_1\) and \(m_2\) which gives \(3/5 * -5/3 = -1\). Since the product equals -1, the two lines are perpendicular.

Key Concepts

Understanding the Slope of a Line

The slope of a line measures how steep the line is. It's like figuring out how much you go up or down as you move from one point to another along the line. If you have two points,

• Call them \((x_1, y_1)\) and \((x_2, y_2)\).
• The formula to find the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• For example, if your points are \((-5, -1)\) and \((0, 2)\),
• You plug them into the formula:\(m_1 = \frac{2 - (-1)}{0 - (-5)} = \frac{3}{5}\).
• This tells you that for every 5 steps you move to the right, you go up 3 steps.
  Remember:
  • Positive slope means the line goes up.
  • Negative slope means it goes down.
  • Zero slope means it's a flat line.
  • Undefined slope means it's a vertical line.

The Product of Slopes and Perpendicular Lines

Two lines are said to be perpendicular if they intersect at a right angle. To see if two lines are perpendicular using their slopes, you look at the product of their slopes:

• If the product is -1, then the lines are perpendicular.
• For example, we calculated the slopes of two lines:\(m_1 = \frac{3}{5}\)and\(m_2 = -\frac{5}{3}\).You multiply these:\(m_1 \times m_2 = \frac{3}{5} \times -\frac{5}{3} = -1\).This confirms that the two lines are perpendicular.
  The key points are:
  • The slopes must multiply to -1.
  • It reflects that the lines create a 90-degree angle.

Formulating Algebraic Equations from Slopes

Algebraic equations are like recipes for lines on a graph. You can use the slope along with a point to write an equation for a line. The basic form of a linear equation is the slope-intercept form:\(y = mx + b\), where:

• \(m\) is the slope.
• \(b\) is the y-intercept, or where the line crosses the y-axis.
• To write the equation of a line through \((-5, -1)\) with\(m = \frac{3}{5}\), you plug these into the equation form:\(-1 = \frac{3}{5}(-5) + b\).
• Solve for \(b\) to find:\(-1 = -3 + b\) which simplifies to \(b = 2\).
• Now, the equation is:\(y = \frac{3}{5}x + 2\).
  This form lets you graph or predict other points on the line easily.
  Understanding this helps in drawing lines and points that fit the line's equation correctly.