Problem 55 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (-3,-4),(1,-1) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 4 = \frac{3}{4}(x + 3)\).

Step 1: Calculate the Slope

Calculate the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the coordinates of the two given points into the formula: \(m = \frac{-1 - (-4)}{1 - (-3)} = \frac{3}{4}\).

Step 2: Writing the Equation

Write the equation in point-slope form using the slope calculated in step 1 and the coordinates of one of the given points. For example, using the point (-3, -4), the equation becomes: \(y - (-4) = \frac{3}{4}(x - (-3))\)

Step 3: Simplifying the Equation

Simplify the equation derived in Step 2 to its simplest form: \(y + 4 = \frac{3}{4}(x + 3)\). This is the equation of the line in point-slope form that passes through the given points.

Key Concepts

Slope Calculation

Calculating the slope of a line is a fundamental step in understanding linear equations. The slope indicates the steepness and direction of a line. To find the slope between two points, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

This formula essentially measures the change in the \( y \)-values over the change in the \( x \)-values. Let's break this down:

• Change in \( y \): Subtract the first \( y \)-coordinate from the second. This gives you how much the line rises or falls.
• Change in \( x \): Subtract the first \( x \)-coordinate from the second. This provides the horizontal distance between the points.

By dividing the change in \( y \) by the change in \( x \), you get the slope \( m \). For instance, between the points \((-3, -4)\) and \((1, -1)\), the slope is \( \frac{3}{4} \), meaning for every 4 units you move right, you go up 3 units.

Equation of a Line

Once you have the slope of a line, you can write its equation in point-slope form. This is a simple and versatile way to express the equation of a line. The formula for point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line.

To create the equation of the line through the points \((-3, -4)\), using a slope of \( \frac{3}{4} \) calculated earlier, simply plug them into the point-slope formula:

• Choose a point: Here, \(-3, -4\) works well.
• Insert the slope: Use the \( \frac{3}{4} \).

Insert these into the formula to get \( y + 4 = \frac{3}{4}(x + 3) \).

This form is particularly helpful because it's straightforward to derive it once you know a point and the slope.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate plane to explore geometric concepts. It ties together algebra and geometry through the use of coordinates (\(x, y\)). It is instrumental in deriving critical information about lines, such as length, midpoint, and slope.

This subject allows for a deep investigation of how geometric shapes behave in the plane. For example:

• Lines: Defined by their slopes and points.
• Circles and curves: Analyzed for their radii or focal points.

By plotting points and examining their relationships, coordinate geometry provides a powerful framework for solving real-world problems through mathematical models.