Problem 28 - Algebra 2

Question

Write in point-slope form the equation of the line through each pair of points. \((-4,10)\) and \((-6,15)\)

Step-by-Step Solution

Answer

The point-slope form of the line passing through the points (-4,10) and (-6,15) is: y - 10 = (-5/2)(x + 4).

Step 1: Calculate the slope of the line

To find the slope (m) of the line passing through the two points (-4, 10) and (-6, 15), use the slope formula:m = (y2 - y1) / (x2 - x1).Substitute the coordinates of the points into the formula:m = (15 - 10) / (-6 + 4) = 5 / (-2) = -5/2.

Step 2: Write the point-slope form of the equation

The point-slope form of the equation of a line is given by:y - y1 = m(x - x1).Choose one of the points to use, for example (-4,10). Substitute the slope and the coordinates of this point into the equation:y - 10 = (-5/2)(x - (-4)).Simplifying, we get the point-slope form:y - 10 = (-5/2)(x + 4).

Key Concepts

Slope Calculation

Understanding the concept of slope is essential when dealing with linear equations. The slope is a measure of the steepness or angle of a line, often symbolized as 'm'. In algebra, the slope can be calculated from any two points on a line using the formula: \[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1} \].

When we calculate the slope between two points, say \( (-4,10) \) and \( (-6,15) \) as in our exercise, we plug the coordinates into our slope formula: \[ m = \frac{15 - 10}{-6 - (-4)} = \frac{5}{-2} = -\frac{5}{2} \].

This result tells us several things about the line: for every two units that we move horizontally to the right (the run), the line falls by five units vertically (the rise). This negative slope indicates that the line is descending from left to right. Grasping the sign and the magnitude of the slope helps in visualizing the orientation of the line on a graph.

Linear Equations

Linear equations form the backbone of algebra and are equations that make straight lines when graphed on the coordinate plane. These equations typically are expressed in various forms like the standard form \( Ax + By = C \) and the slope-intercept form \( y = mx + b \) where 'm' represents the slope and 'b' the y-intercept. A more versatile form, particularly when working with specific points on a line, is the point-slope form equation: \[ y - y_1 = m(x - x_1) \].

This equation establishes a relationship involving the slope (\( m \) - steepness of the line) and a point (\( (x_1, y_1) \) - specific location on the line). Applying the point-slope form, once you discern the slope and have a point on the line, constructing the equation is straightforward:

• Insert the slope for 'm'.
• Put in the x and y values of your specific point for \( x_1 \) and \( y_1 \) respectively.

Returning to our example, with a calculated slope (\( m = -\frac{5}{2} \) ) and a chosen point (-4, 10), the point-slope equation becomes:\[ y - 10 = -\frac{5}{2}(x + 4) \].

Algebraic Expressions

At the core of algebra, algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (e.g., addition, subtraction, multiplication, division). These expressions become incredibly useful for representing real-world situations and mathematical relationships in a concise and general form.

In an equation, algebraic expressions can appear on either side of an equal sign. When we derived the point-slope form equation for our linear equation, we manipulated algebraic expressions to arrive at \[ y - 10 = -\frac{5}{2}(x + 4) \]. This result is a very specific algebraic expression: it conveys valuable information about the line's slope and includes variables \( x \) and \( y \) that give the line its functionality across the infinite points it passes through.

Understanding algebraic expressions and how to manipulate them is critical for solving equations. Algebraic techniques such as distributing, factoring, and combining like terms are used to simplify expressions and solve for variables, enabling students to unravel more complex problems involving this area of mathematics.