Problem 94 - College Algebra

Question

Then use the TRACE feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=-\frac{1}{2} x-5$$

Step-by-Step Solution

Answer

The calculated slope of the line using two points P(-4, -3) and Q(2, -6) is -0.5, which matches with the coefficient of \(x\) (-0.5) in the given line's equation \(y=-\frac{1}{2} x-5\). Thus, we can conclude that the calculated slope is accurate.

Step 1: Select Points on the Line

Choose two points via the TRACE feature. Let's say the points are P(-4, -3) and Q(2, -6).

Step 2: Calculate the Line's Slope

The slope \(m\) of a line can be calculated using the coordinates of two points on the line with the formula: \(m = \frac{y_2-y_1}{x_2-x_1}\). Therefore, \(m = \frac{-6-(-3)}{2-(-4)} = \frac{-3}{6}= -0.5\). The slope of the line is -0.5.

Step 3: Compare the Slope with the Coefficient of \(x\)

The coefficient of \(x\) in the line's equation is -0.5. The calculated slope also happens to be -0.5, implying that the calculation is correct.

Key Concepts

Coordinate Geometry

In the world of mathematics, coordinate geometry, also known as analytic geometry, merges algebra and geometry using a coordinated plane or coordinate system. Two-dimensional space often features the renowned Cartesian plane, helping us locate points and create geometric shapes using coordinates.
When you work with coordinate geometry, you aim to understand and describe figures using an ordered pair \( (x, y) \). The \( x \) value represents the horizontal placement, while the \( y \) value indicates how high or low the point is placed on the vertical axis.

• The Cartesian plane consists of a horizontal axis (often called \( x \)-axis) and a vertical axis (often called \( y \)-axis).
• Every point can be depicted as \( (x, y) \), meaning "x" units across and "y" units up or down.

Simply put, coordinate geometry equips you with precise mathematical techniques for describing geometrical figures. Once you're familiar with it, you'll find it easier to handle concepts like lines, angles, and figures in the plane.

Linear Equations

A linear equation in two variables is most often presented in the slope-intercept form \(y = mx + b\). Here, \(y\) and \(x\) are the variables, \(m\) represents the slope, and \(b\) is the y-intercept, the point where the line crosses the \(y\)-axis.
Linear equations are fundamental in constructing lines on a graph. They provide a straightforward method for mapping out points and understanding the relationship between variables.
A linear equation manifests as a straight line when plotted on a graph, distinguishing it from curves like parabolas or circular shapes.

• The slope \(m\) indicates how steep or flat the line is.
• The y-intercept \(b\) tells you where the line intersects with the y-axis.
• Straight lines are characterized by a constant rate of change, reflected by the slope.

Recognizing these components can make linear equations much easier to understand and work with, especially when transitioning to more complex mathematical concepts.

Slope Formula

The slope formula is quintessential to coordinate geometry and involves determining the steepness or incline of a line connecting two points on a plane. Given two points \(P(x_1, y_1) \) and \(Q(x_2, y_2) \), the slope \(m\) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula gives a ratio explaining how much the \(y\) coordinate changes for a change in the \(x\) coordinate, forming the foundation for understanding gradients.

• If \(m > 0\), the line ascends as it moves from left to right.
• If \(m < 0\), the line descends as it moves from left to right.
• If \(m = 0\), the line is horizontal, indicating no vertical change.

Using this formula, you can evaluate the slope of a line effectively, whether for algebraic problems like the one in this exercise or for geometric analysis. Understanding how to calculate and interpret the slope is crucial to mastering concepts in both algebra and geometry.