Problem 24 - 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. $$ (1,9),(9,9) ; x=1 $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (1,9) and (9,9) is \(y = 9\). The lines \(y = 9\) and \(x = 1\) are perpendicular to each other. This is confirmed by a graphical representation.

Step 1: Find the Equation of the Line Through the Points

The formula of the slope for two given points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). The given points are (1,9) and (9,9). So, the slope \(m = \frac{{9 - 9}}{{9 - 1}} = 0\). The intercept form of the equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the line passes through the points (1,9), so if we put \(x=1, y=9\), and \(m=0\) in the equation, we can solve for \(b\). Doing that, we get \(9 = 0*1 + b\), so \(b = 9\). So the equation of the line is \(y = 0*x + 9\), which simplifies to \(y = 9\).

Step 2: Show the Line is Perpendicular to the Given Line

Two lines are perpendicular if the product of their slopes is -1. The given line is \(x=1\), which is a vertical line and its slope is undefined. The slope of the line we found in step 1 is 0. The product of an undefined value (which can be thought of as infinity) and zero is undefined, but in this context, we say it is -1. So, the lines are indeed perpendicular.

Step 3: Graph the Lines

Both lines can be graphed on a coordinate plane. Line 1: \(y = 9\) is a horizontal line running through the point where \(y = 9\). Line 2: \(x = 1\) is a vertical line running through the point where \(x = 1\). As expected, the lines intersect at right angles, which confirms they are perpendicular.

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in algebra and geometry. It represents all the possible points that a straight line contains on a coordinate plane. The most commonly used form for the equation of a line is the slope-intercept form, given by:\[ y = mx + b \]where:

• \( m \) is the slope of the line, which tells us how steep the line is.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

In our specific case, the line passes through two points (1,9) and (9,9). We first calculate the slope \( m \) using the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For our points, this becomes:\[ m = \frac{9 - 9}{9 - 1} = \frac{0}{8} = 0 \]A slope of 0 means the line is perfectly horizontal. Plugging this into the slope-intercept form gives us the equation of the line:\[ y = 0x + 9 = 9 \]This equation indicates a horizontal line where \( y \) is always 9, regardless of the \( x \) value.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which is 90 degrees. The concept of perpendicularity in terms of slopes is that their slopes multiply to give -1, except when dealing with vertical or horizontal lines.Vertical lines have an "undefined" slope, while horizontal lines have a slope of 0. When examining whether the line \( y = 9 \) (horizontal) is perpendicular to the line \( x = 1 \) (vertical):

• The product of an undefined slope and zero is not a straightforward calculation.
• However, by geometric interpretation, a vertical and a horizontal line are inherently perpendicular.

Therefore, even though the slope of the vertical line is undefined, these two lines \((y = 9 \text{ and } x = 1)\) are indeed perpendicular, meeting at a right angle.

Graphing Lines

Graphing lines effectively helps to visualize the relationships and intersections between different lines on a plane. Let’s graph our lines from the exercise:### Horizontal Line: \( y = 9 \)- This line runs parallel to the x-axis and crosses the y-axis at \( y = 9 \).- Every point on this line has a y-coordinate of 9.### Vertical Line: \( x = 1 \)- This line runs parallel to the y-axis and crosses the x-axis at \( x = 1 \).- Every point on this line has an x-coordinate of 1.When these two lines are plotted on the same graph, they should intersect at the point (1,9), where \( x = 1 \) meets \( y = 9 \).
This intersection forms a right angle, confirming that these lines are perpendicular as checked algebraically. Graphing is a powerful tool to validate algebraic calculations visually and helps in understanding how different elements on a coordinate plane interact.