Problem 25 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given points. $$ (-5,10),(-4,-2) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (-5,10) and (-4,-2) is \(y - 10 = -12x - 60\).

Step 1: Calculate the Slope

The first step is to calculate the slope (m) of the line using the slope formula for the points order (x_1, y_1), and (x_2, y_2): \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Here \(x_1 = -5\), \(y_1 = 10\), \(x_2 = -4\), and \(y_2 = -2\). Thus, \(m = \frac{{-2 - 10}}{{-4 - (-5)}} = \frac{{-12}}{{1}} = -12 \).

Step 2: Apply Point-Slope Form

Next, you will apply the point-slope form of a linear equation which is \(y - y_1 = m(x - x_1)\). Substituting the value of m, which is -12 and the coordinates of one of the points (-5,10) as x_1 and y_1, the equation becomes \(y - 10 = -12(x - (-5))\).

Step 3: Simplify the Equation

Finally, the equation \(y - 10 = -12(x + 5)\) should be simplified. This gives the final equation as \(y - 10 = -12x - 60\).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics, especially when you begin exploring algebra. Essentially, they describe a straight line and are the simplest type of equations that involve variables. A linear equation typically looks like this: \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

In the context of the exercise you are working on, the line needs to pass through two given points. Linear equations can be expressed in various forms, such as point-slope form or slope-intercept form. The advantage of using the point-slope form, as in this problem, is that it makes it very clear how the slope and a particular point from a line contribute to the overall equation.

When solving problems like these, being able to identify the right form and convert between different forms of linear equations is a crucial skill for success.

Slope Calculation

Calculating the slope of a line is a fundamental part of many geometry problems, particularly those involving linear equations. The slope tells us how steep the line is. It's calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This captures the rate of change between the two points, \((x_1, y_1)\) and \((x_2, y_2)\).

In the given exercise, we saw two points: \((-5, 10)\) and \((-4, -2)\). By substituting these points into the slope formula, we found that the slope \(m\) is \(-12\).

Understanding the slope is essential because it provides valuable information about the direction and angle of the line.

• A positive slope means the line rises as it moves to the right.
• A negative slope indicates the line falls as it moves to the right.
• A zero slope results in a horizontal line.
• Undefined slope corresponds to a vertical line.

Grasping these concepts not only aids in solving these problems but also enhances your overall understanding of geometry.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using an algebraic approach to solve geometrical problems. It is a powerful method because it allows precise calculation and visualization of different geometrical shapes and lines.

Here, we are using coordinate geometry principles to determine the equation of a line that passes through two given points. By plotting the points \((-5, 10)\) and \((-4, -2)\) on the Cartesian plane, you can visualize the line that connects them. This visualization is greatly aided by knowing the slope and the equation of the line, which is derived using point-slope form in this case.

Coordinate geometry allows for a concrete understanding of abstract mathematical principles. It makes it easy to see how changes in equations affect the position and direction of lines in a plane.

By mastering coordinate geometry, students can better understand more complex mathematical topics and apply these skills in various practical contexts, such as physics, engineering, and computer graphics.