Problem 44 - Introductory Algebra for College Students

Question

The line passing through \((-2, y)\) and \((-4,4)\) is perpendicular to the line passing through \((-1,-2)\) and \((4,-1)\) Find \(y\)

Step-by-Step Solution

Answer

The value of \(y\) that will make the lines perpendicular is \(y = 18/5\).

Step 1: Determine the slope of the first line

Calculate the slope of the first line using the formula \((y2 - y1) / (x2 - x1)\). In this case, \(x1 = -2\), \(x2 = -4\), \(y1 = y\), and \(y2 = 4\). This gives a slope \(m1\) of \((4-y) / (-4 - -2)\), or \((4-y) / -2\). So, \(m1 = -1/2 * (4-y)\). This simplifies to \(m1 = (y - 4) / 2\).

Step 2: Determine the slope of the second line

The slope of the second line, \(m2\), can be calculated in a similar way, using the coordinates given: \((-1,-2)\) and \((4,-1)\). Plugging into the formula gives \((-1 - -2)/(4 - -1)\), which simplifies to \(m2 = 1/5\).

Step 3: Set the slopes to be perpendicular and solve for \(y\)

Since two lines are perpendicular if the product of their slopes is -1, we set \((y - 4)/2 * 1/5 = -1.\) Solving for \(y\) gives \(5*(y - 4)/2 = -1\), simplifying to \(5y - 20 = -2\), and finally \(y = (-2+20)/5\).

Key Concepts

Slope of a Line

To understand the idea of the slope of a line in coordinate geometry, imagine it as a measure of how steep the line is. The slope is a quantitative value that describes this steepness. It indicates how much the line "climbs" vertically when you move a certain distance horizontally.

For instance, a slope can tell you how much the "rise" is over the "run" between two points on a line. A positive slope tells us that the line is rising as we move from left to right, while a negative slope tells us the line is falling.

• If the slope is zero, the line is perfectly horizontal.
• If the slope is undefined, the line is vertical.

Understanding the slope of a line is vital because it helps us analyze and predict the behavior of linear relationships.

Slope Formula

The slope formula is a key tool in coordinate geometry. It provides a way to calculate the slope using two points on a line. The general formula is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. This formula computes the change in the y-values (vertical change) divided by the change in the x-values (horizontal change) between these points.

• If points are very close, the formula will still provide the slope accurately.
• If the line is horizontal, \(y_2 - y_1\) will be zero, resulting in a slope of zero.
• If the line is vertical, \(x_2 - x_1\) will be zero, making the slope undefined.

Using the slope formula, we can analyze and interpret the relationships between various points on a line effectively.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, deals with geometrical shapes in a coordinate plane. The basic idea is to use ordered pairs of numbers, \(x, y\), to determine the position of points on the plane.

In problems dealing with lines, like the exercise provided, you use coordinate geometry to:

• Locate points using coordinates.
• Calculate distances between points and slopes of lines.
• Analyze the properties of geometric figures.

Coordinate geometry provides a powerful way to connect algebraic equations with geometric figures. You can determine many properties of a shape just by analyzing its coordinates and equations.

Linear Equations

Linear equations in two variables, such as \(y = mx + b\), describe straight lines on a coordinate plane. In this equation, \(m\) represents the slope, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the context of this exercise, once we know the slopes, we set the product of their slopes to equal \(-1\) for perpendicular lines. This relationship is crucial:

• Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1 \cdot m_2 = -1\).
• Understanding this allows you to solve for variables, like \(y\) in such geometric problems.

Linear equations provide a systematic way to describe the behavior and relationship of straight lines through their slopes and intercepts.