Problem 61 - Introductory Algebra for College Students

Question

Write the slope-intercept form of the equation of the Tine passing through \((-5,6)\) and \((3,-10)\)

Step-by-Step Solution

Answer

The slope-intercept form for the line passing through (-5,6) and (3,-10) is \(y = -2x - 4\).

Step 1: Find the Slope of the Line

We use the formula \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\). Here, \(x_1 = -5, y_1 = 6, x_2 = 3, y_2 = -10\). So, \(m = \frac{-10 - 6}{3 - (-5)} = \frac{-16}{8} = -2\).

Step 2: Substitute the Slope and One Point into the Equation

Now we substitute the slope and one of the points (-5,6) in the equation of the line \(y = mx + b\), this gives us \(6 = -2 * -5 + b = 10 + b\).

Step 3: Solve for the Y-intercept

Next, we rearrange the equation to solve for \(b\): \(6 = 10 + b\) leads to \(b = 6 - 10 = -4\).

Key Concepts

Equation of a Line

The equation of a line in its most commonly used form is called the slope-intercept form. This form makes it easy to identify two key features of a straight line: the slope and the y-intercept. The slope-intercept form of the equation of a line is written as:

• \(y = mx + b\)

In this equation:

• \(y\) represents the dependent variable, typically plotted on the y-axis of a graph.
• \(x\) stands for the independent variable, plotted on the x-axis.
• \(m\) is the slope of the line, indicating how steep the line is.
• \(b\) is the y-intercept, where the line crosses the y-axis.

Using this form allows for quick and easy graphing, as long as you know the slope and y-intercept of the line. This familiarity is critical in coordinate geometry, helping us to graphically represent the relationships between variables.

Slope Calculation

The slope of a line represents the rate of change between any two points on that line. It also tells us how steep a line is. To find the slope, use the formula:

• \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\)

Here's how to calculate it:

• Select two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the y-coordinates: \(y_2 - y_1\).
• Subtract the x-coordinates: \(x_2 - x_1\).
• Divide the difference in y by the difference in x to find the slope \(m\).

In our exercise, the points \((-5,6)\) and \((3,-10)\) were used, resulting in a slope \(m = -2\). The negative slope indicates the line decreases in value as it moves from left to right.

Y-intercept

The y-intercept is where a line crosses the y-axis on a graph. This point is crucial because it provides a starting value when \(x = 0\).

• Substitute the slope \(m\) and one of the known points into the slope-intercept form \(y = mx + b\).
• For example, with \(m = -2\) and the point \((-5,6)\): \(6 = -2(-5) + b\).
• Solve for \(b\): \(6 = 10 + b\), so \(b = -4\).

This calculation shows that the line crosses the y-axis at \(b = -4\). With both the slope and y-intercept found, you have all the components to write the complete equation of the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry using a coordinate system. This method allows for the precise description of geometric figures and their relationships.

• Lines, curves, and polygons can be expressed using equations.
• The Cartesian coordinate system plots points using ordered pairs (x, y), enabling analysis of geometric shapes.
• Through coordinate geometry, we translate geometric problems into algebraic equations, facilitating easier problem-solving.

For example, knowing two points' coordinates enables calculation of a line's slope and equation, demonstrating a powerful tool for solving complex problems involving shapes and distances.