Problem 23 - College Algebra

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. $$ (1,2) \text { with a slope of }-\frac{4}{5} $$

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{4}{5}x + \frac{14}{5}\).

Step 1: Identify Point and Slope

We start by identifying the given point and the slope of the line. The point provided is \((1, 2)\) and the slope \(m\) is \(-\frac{4}{5}\).

Step 2: Apply Point-Slope Formula

The point-slope form of a line is given by the equation \(y - y_1 = m(x - x_1)\). Substituting the point \((1, 2)\) and the slope \(-\frac{4}{5}\), the equation becomes: \[y - 2 = -\frac{4}{5}(x - 1)\]

Step 3: Expand the Equation

Distribute \(-\frac{4}{5}\) on the right side of the equation: \[y - 2 = -\frac{4}{5}x + \frac{4}{5}\]

Step 4: Solve for y (Convert to Slope-Intercept Form)

Add 2 to both sides to solve for \(y\): \[y = -\frac{4}{5}x + \frac{4}{5} + 2\]Convert 2 into a fraction with a denominator of 5: \[y = -\frac{4}{5}x + \frac{4}{5} + \frac{10}{5}\]Combine the fractions: \[y = -\frac{4}{5}x + \frac{14}{5}\]

Step 5: State the Final Equation in Slope-Intercept Form

The equation of the line in slope-intercept form (\(y = mx + b\)) is: \[y = -\frac{4}{5}x + \frac{14}{5}\]

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is widely used because it provides direct information about the line's slope and y-intercept. This form is represented as \(y = mx + b\), where \(m\) is the slope of the line, indicating how steep the line is, and \(b\) is the y-intercept, representing the point where the line crosses the y-axis.
Understanding this form can make it easier to graph linear equations or analyze the relationship between two variables in coordinate geometry.

• When \(b = 0\), the line passes through the origin, which is the point (0, 0). In this case, the equation simplifies to \(y = mx\).
• If \(m = 0\), the line is horizontal, and the equation becomes \(y = b\).

In the context of our exercise, we practiced transforming a point-slope form into the slope-intercept form to identify the specific slope and y-intercept for the given line.

Linear Equations

Linear equations are mathematical expressions that model a straight line when graphed on a coordinate plane. These equations can have different forms, but they all describe the relationship between two variables. Linear equations can determine a variety of real-world situations, including calculating rates of change and predicting future outcomes.
When working with linear equations, the key goals are often to:

• Find the slope, which tells us the direction and steepness of the line.
• Identify the y-intercept, which signifies where the line crosses the y-axis.
• Convert the equation into different forms for easier interpretation, such as from point-slope to slope-intercept form.

This process is evident in our example, where we initially used the point-slope form based on given data, then transformed it to understand the line's complete equation in slope-intercept form.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry to study points, lines, and figures using coordinate systems. It mainly involves the representation and analysis of geometric figures through algebraic equations, helping solve complex geometric problems with clarity and precision.
Several key concepts in coordinate geometry include:

• Points: Defined by coordinates in the form \((x, y)\) where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance.
• Lines: Described by linear equations such as \(y = mx + b\), showcasing the relationship between coordinates on a plane.
• Slopes: Indicating the inclination or steepness of a line, calculated by \(\frac{\text{change in } y}{\text{change in } x}\).

The task in the example uncovers the practical use of coordinate geometry principles, as we started with a specific point and slope to derive the equation of a line and understand its characteristics and graphical representation better.