Problem 21 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line passing through the two points. Show that the line is perpendicular to the given line. Check your answer by graphing both lines. $$ (-2,-2),(1,-3) ; y=3 x-1 $$

Step-by-Step Solution

Answer

The equation of the line is \(y = -1/3x - 8/3\). It is verified that this line is perpendicular to the line \(y = 3x - 1\). The verification on the graph should reveal that the lines intersect at a right angle.

Step 1: Find the Slope

The first step is to find the slope of the line passing through the points \((-2,-2)\) and \((1,-3)\). It can be found using the formula for slope which is \((y_2 - y_1) / (x_2 - x_1)\), so substituting the points gives \(m = (-3 - (-2)) / (1 - (-2)) = -1 / 3\)

Step 2: Find the y-intercept

Slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Since we know slope \(m = -1 / 3\), we can substitute one of the points and the slope into the equation to solve for \(b\). Let's use the point \((-2,-2)\), giving \( -2 = -1/3 * -2 + b\), hence \(b = -2 - 2/3 = -8/3\)

Step 3: Verify Perpendicularity

Two lines are perpendicular if the product of their slopes is -1. The slope of the given line \(y = 3x - 1\) is 3. So multiplying the slopes of the obtained line and the given line, we get \((-1/3) * 3 = -1\), confirming that the lines are indeed perpendicular.

Step 4: Graph the Lines

To graph the lines, first, draw the line for \(y = 3x - 1\) using the slope and y-intercept. Then draw the line for \(y = -1/3x - 8/3\), also using the slope and y-intercept. The lines should intersect at a right angle, further confirming they are perpendicular. The actual graphing is an exercise left for the student, though, as it is impossible to graph within this text-based system.

Key Concepts

Understanding Perpendicular Lines

Perpendicular lines are quite special in geometry. When two lines are perpendicular, they intersect at a right angle, which is 90 degrees. This is a distinctive feature and it helps us understand their relationship on a graph. To check if two lines are perpendicular, we use their slopes.

• If you multiply the slopes of two lines and the product is -1, the lines are perpendicular.
• In our problem, the slope of one line is \(3\) and the other line has a slope of \(-\frac{1}{3}\). When we multiply these slopes, we get \(-1\) which confirms perpendicularity.

Understanding the concept of perpendicular lines is crucial because it helps you solve many problems in coordinate geometry. Always remember, perpendicularity and slope go hand in hand, and checking the slope product is a quick way to confirm it.

Linear Equations in Slope-Intercept Form

Linear equations can be written in a special way called the \'slope-intercept form\'. This form is very useful:

• Slope-intercept form is \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.
• The slope \(m\) describes the steepness of the line, while the y-intercept \(b\) is the point where the line crosses the y-axis.

In the given exercise, we calculated the slope using two points. Then, we used this slope to find the y-intercept by substituting one of the points into the equation: \[y = -\frac{1}{3}x - \frac{8}{3}\]This form is particularly useful for graphing because it directly shows how to start plotting the line, using these two parameters.

Graphing Linear Equations

Graphing is an excellent way to visualize linear equations. It helps to see the relationship between different expressions and conditions like perpendicularity.

• Start by plotting the y-intercept on the graph which is the point where the line crosses the y-axis.
• Use the slope to determine the direction and steepness of the line. For example, a slope of \(3\) means that for every unit increase in 'x', 'y' increases by 3 units. Conversely, \(-\frac{1}{3}\) means for every unit increase in 'x', 'y' decreases by \(-\frac{1}{3}\).

By graphing the lines from your equations, \(y = 3x - 1\,\) and \(y = -\frac{1}{3}x - \frac{8}{3}\,\) you'll visually confirm their perpendicular nature as the lines intersect at a right angle. Always remember graphing can illuminate many relationships and checks, making it a powerful tool in geometry.