Problem 7 - Algebra 1: Concepts and Skills

Question

Write the equation of the line passing through the two points. Show that this line is perpendicular to the given line. $$ (-3,0),(3,6) ; y=-x-2 $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (-3,0) and (3,6) is \(y = x + 3\). This line is perpendicular to the line \(y = -x - 2\).

Step 1: Calculate the Slope

The slope (m1) of the line passing through the points \((-3,0)\) and \((3,6)\) can be calculated using the formula: \(m1 = (y_2 - y_1) / (x_2 - x_1)\). That gives us \(m1 = (6 - 0) / (3 - -3) = 6/6 = 1\).

Step 2: Determine the Line Equation

The point-slope form of the line equation is \(y - y_1 = m1(x - x_1)\). By substituting \(m1 = 1\) and using the point \((-3,0)\), we get the equation of the line as \(y - 0 = 1(x - -3)\), which simplifies to \(y = x+3\).

Step 3: Test for Perpendicular Lines

The given line has equation \(y = -x - 2\). It can be seen that the slope (m2) of this line is -1. To check if two lines are perpendicular, the product of their slopes should be -1. Therefore, \(m1 * m2 = 1 * -1 = -1\), verifying that the two lines are in fact perpendicular.

Key Concepts

Equation of a Line

Finding the equation of a line involves determining how the line behaves on a coordinate plane. To write the equation, we often use forms such as slope-intercept form or point-slope form.
In many cases, especially when we have a point and a slope, the point-slope form is very useful. It looks like this: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.

• For our exercise, we started with the coordinates \((-3,0)\) and a calculated slope of 1. Using these in the point-slope formula, we derived the line equation \(y = x + 3\).
• The equation tells us that the line crosses the y-axis at 3, which is the y-intercept. The slope of 1 indicates that for each unit increase in \(x\), \(y\) also increases by 1.
• Converting to slope-intercept form (\(y = mx + b\)) can reveal the intercept directly, making it easier to graph.

Perpendicular Lines

Lines are considered perpendicular when they intersect at a right angle (90 degrees). This relationship between the two lines can be detected with the help of their slopes.

• The essential condition for perpendicularity is that the product of their slopes is -1. This stems from the geometric concept that the slopes are negative reciprocals of each other.
• In our exercise, the original line equation given was \(y = -x - 2\), where the slope is \(-1\). Our calculated slope from the given points was \(1\), and their product \(1 \times -1 = -1\) confirms perpendicularity.
• Going forward, you only need to calculate the slopes of the two lines. If their product is -1, those lines will certainly be perpendicular.

Slope Calculation

The slope of a line expresses its steepness and direction. You calculate the slope using two points on the line with the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This calculation gives insight into how much \(y\) changes with a change in \(x\).

• For the points \((-3,0)\) and \((3,6)\), the slope calculated is \(1\), meaning the line is rising at a 45-degree angle when plotted.
• In general, a positive slope means the line rises towards the right, while a negative slope implies it falls. A zero slope indicates a horizontal line, and an undefined slope (where \(x_2 = x_1\)) is vertical.
• Knowing how to calculate and interpret slope is foundational for understanding more complex geometric concepts in algebra and beyond.