Problem 11 - Calculus Single Variable

Question

Write the point-slope equation of the line determined by the two given points. (-7,-4),(9,0)

Step-by-Step Solution

Answer

The point-slope equation is \( y + 4 = \frac{1}{4}(x + 7) \).

Step 1: Identify the Point-Slope Formula

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope of the line, and \((x_1, y_1)\) is a point on the line.

Step 2: Calculate the Slope

To find the slope \( m \), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plug in the coordinates of the points: \((-7, -4)\) and \((9, 0)\). So, \( m = \frac{0 - (-4)}{9 - (-7)} = \frac{4}{16} = \frac{1}{4} \). The slope is \( m = \frac{1}{4} \).

Step 3: Choose One Point

You can choose either of the two points to use in the point-slope formula. For this solution, we will use the point \((-7, -4)\).

Step 4: Substitute Into the Point-Slope Formula

Now, use the point \((-7, -4)\) and the slope \( m = \frac{1}{4} \) in the point-slope formula: \( y - (-4) = \frac{1}{4}(x - (-7)) \). This simplifies to \( y + 4 = \frac{1}{4}(x + 7) \).

Step 5: Simplify the Equation (Optional)

If required, you can simplify the equation further by distributing the slope and isolating \( y \): \( y + 4 = \frac{1}{4}x + \frac{7}{4} \). To isolate \( y \), subtract 4 from both sides: \( y = \frac{1}{4}x + \frac{7}{4} - 4 \) which simplifies to \( y = \frac{1}{4}x - \frac{9}{4} \).

Key Concepts

Slope Calculation

The concept of slope calculation is essential when you need to describe the steepness or incline of a line. It is determined by comparing the vertical change (rise) over the horizontal change (run) between two coordinate points on a line. The formula used to calculate the slope \( m \) is:

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

Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points.
For instance, if you have points \((-7, -4)\) and \((9, 0)\), you substitute these values into the formula to determine the slope:

• Subtract the y-values: \(0 - (-4) \) gives you 4.
• Subtract the x-values: \(9 - (-7) \) gives you 16.
• So, the slope \( m = \frac{4}{16} = \frac{1}{4} \).

Thus, the slope \( \frac{1}{4} \) signifies that for every four units of horizontal movement, the line rises one unit vertically.

Linear Equations

Linear equations are equations that make a straight line when they are graphed on a coordinate plane. They are fundamental in algebra and coordinate geometry as they describe relationships where the rate of change is constant.
There are different forms of linear equations. One of the most common ways to express them is through the point-slope form:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \)

This formula is particularly useful when you have the slope \( m \) and a point \((x_1, y_1)\) on the line. Once you know these, you can easily write the equation for the line.By plugging \((-7, -4)\) and \(m = \frac{1}{4}\) into the point-slope formula, you derive:

• \( y - (-4) = \frac{1}{4}(x - (-7)) \)

Simplifying gives you:

• \( y + 4 = \frac{1}{4}(x + 7) \)

Optional further simplification gives:

• \( y = \frac{1}{4}x - \frac{9}{4} \)

This form is intuitive as you can clearly see how changes in \( x \) influence changes in \( y \).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to describe the position and relationships between points, lines, and shapes on a graph. It uses a system of coordinates, generally the Cartesian coordinate system, which includes a horizontal x-axis and a vertical y-axis.In this system:

• A point is represented by a pair of numbers, \((x, y)\), that indicate its location on the plane.
• Lines can be defined via equations such as the slope-intercept form, point-slope form, or standard form.
• For a line passing through the points \((-7, -4)\) and \((9, 0)\), coordinate geometry enables us to identify and plot these points accurately and derive their relationship using equations.

In the context of linear equations, coordinate geometry provides a clear and visual way of understanding how lines interact and behave in space. By having a visual grasp of these concepts, students can better understand how algebraic equations map onto geometric figures. This forms a basis for advanced studies in mathematics and helps solve practical problems in fields like engineering, physics, and computer science.