Problem 67 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=\frac{2}{3} x-\frac{1}{2}$$

Step-by-Step Solution

Answer

The points are (0, -1/2) and (3, 3/2); the slope is 2/3.

Step 1: Identify Two Points on the Line

To find two points on the line represented by the equation \(y=\frac{2}{3}x-\frac{1}{2}\), we assign values to \(x\) and solve for \(y\). Start by choosing \(x=0\): \[y=\frac{2}{3}(0)-\frac{1}{2} = -\frac{1}{2}\]So, the first point is \((0, -\frac{1}{2})\).

Step 2: Find a Second Point

Choose a second value for \(x\) and solve for \(y\). Let \(x=3\): \[y = \frac{2}{3}(3) - \frac{1}{2} = 2 - \frac{1}{2} = \frac{3}{2}\]So, the second point is \((3, \frac{3}{2})\).

Step 3: Use the Points to Find the Slope

The slope \(m\) of the line can be calculated using the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the points \((0, -\frac{1}{2})\) and \((3, \frac{3}{2})\):\[ m = \frac{\frac{3}{2} - (-\frac{1}{2})}{3 - 0} = \frac{\frac{3}{2} + \frac{1}{2}}{3} = \frac{\frac{4}{2}}{3} = \frac{2}{3}\]

Step 4: Conclusion

The slope of the line, confirmed by both calculation and the original equation, is \(\frac{2}{3}\), and it matches the coefficient of \(x\) in the line equation \(y = \frac{2}{3}x - \frac{1}{2}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this system, each point can be uniquely identified by an ordered pair of numbers, usually referred to as \((x, y)\). This allows us to use algebraic techniques to solve geometric problems efficiently.In the context of finding the slope of a line, coordinate geometry enables us to compute attributes like distance, midpoint, and slope between two points. When we work with equations of lines, we graph them based on these pairs of numbers, which makes it easier to understand the geometric relationships between points.

• Defining Points: Every point on a graph is described by coordinates \((x, y)\).
• Graphing Lines: A straight line can be uniquely defined by two points on a coordinate plane.
• Slope and Intercepts: The slope is a measure of the steepness of the line, while intercepts show where the line crosses the axes.

Using coordinate geometry not only makes visualizing lines possible but also provides a systematic way to solve complex problems involving lines and their intersections.

Linear Equations

A linear equation like \(y = \frac{2}{3}x - \frac{1}{2}\) is an equation that makes a straight line when it is graphed. Linear equations express a relationship between two variables usually denoted as \(x\) and \(y\), and are typically written in the form \(y = mx + b\), which is known as the slope-intercept form.

• Slope (\(m\)): This is the rate of change in \(y\) with respect to \(x\). It's the 'rise over run', or how much \(y\) changes for a change in \(x\).
• Y-intercept (\(b\)): This is the point where the line crosses the y-axis. It depicts the value of \(y\) when \(x\) is zero.

To solve for specific points on a line represented by a linear equation, we substitute values for \(x\) to find corresponding \(y\) values (or vice versa). This method is how we identified the points \((0, -\frac{1}{2})\) and \((3, \frac{3}{2})\) in our exercise. Linear equations provide a direct method to model relationships between variables by allowing easy computations of unknown variables.

Point-Slope Formula

The point-slope formula is a valuable tool in coordinate geometry for finding the equation of a line when given a single point on the line and the slope. It is expressed as:\[ y - y_1 = m(x - x_1)\]where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope.Using the point-slope formula is particularly helpful in situations where we readily have a slope from calculations or word problems and a specific point on the line but need to derive the line's equation in slope-intercept form.To apply the point-slope formula:

• Identify a known point \((x_1, y_1)\) on the line.
• Use an already known or calculated slope \(m\).
• Substitute these values into the formula to get the equation of the line.

In this exercise, after finding two points \((0, -\frac{1}{2})\) and \((3, \frac{3}{2})\), we calculated the slope \(m = \frac{2}{3}\). These tools confirm that the line's equation \(y = \frac{2}{3}x - \frac{1}{2}\) is consistent with the slope derived between these points, validating the line's equation and helping in understanding its graphical representation. This method gives us deep insight into algebraic manipulation and basic geometry.