Problem 79 - College Algebra

Question

Find the value of \(y\) if the line through the two given points is to have the indicated slope. \((3, y)\) and \((1,4), m--3\)

Step-by-Step Solution

Answer

The value of y is -2.

Step 1: Identify the known variables

We are given that the slope of the line, \(m\), is -3, the point (1, 4) are \(x_1, y_1\) and the point (3, y) are \(x_2, y_2\). These will be substituted into the formula for slope: \(m = (y2-y1)/(x2-x1)\).

Step 2: Substitute the known variables into the slope formula

Substitute the given values into the formula: -3 = \((y-4)/(3-1)\).

Step 3: Solve for y

Multiply both sides of the equation by the denominator to cross multiply. This will give -6 = y - 4. By adding 4 to both sides to isolate y on one side, you get y = -6 + 4, so y = -2.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a valuable tool in coordinate geometry. It provides an easy way to express the equation of a line when you're given one point on the line and the slope. The general form of the point-slope equation is:

• \( y - y_1 = m(x - x_1) \)

Here,

• \( m \) is the slope of the line.
• \((x_1, y_1)\) is a known point on the line.

Using this form, you can quickly derive the equation of the line by substituting the given values of \( m \), \( x_1 \), and \( y_1 \). This is particularly helpful in problems where you need to find a missing coordinate when other values are specified. Always remember to keep all terms properly substituted and solve systematically.
It simplifies the process, especially when compared to attempting to derive the line equation from scratch. The point-slope form emphasizes the relationship between the slope and any point on the line, effectively anchoring the line's position in the coordinate system.

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, bridges the gap between algebra and geometry by using a coordinate plane to describe geometrical shapes and their properties. In our scenario with the points

• \((1, 4)\)
• \((3, y)\)

and a slope \(m = -3\), coordinate geometry helps us visualize and compute the precise relationship between these points on a grid.
The coordinate plane consists of two perpendicular axes: the x-axis and the y-axis, creating a playground for locating and plotting points. Each point is represented by coordinates \((x, y)\), designating the horizontal and vertical position relative to the origin.
By leveraging the coordinate plane and the concept of slope, we solve for unknown variables, like the \( y \) in our exercise, and fully understand the incline and orientation of lines and shapes present.

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. They take the form \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. A foundational component of algebra, linear equations tell us two essential things:

• The direction or steepness of the line, given by the slope \( m \).
• The point where the line crosses the y-axis, represented by the y-intercept \( b \).

In our example, using the slope formula allowed us to derive an equation that helped solve for an unknown coordinate, \( y \). When given two points like \((3, y)\) and \((1, 4)\) and a slope of \(-3\), the formula easily allowed us to deduce the relationship and ultimately solve for \( y \).
Linear equations are versatile in describing relationships between variables and predicting outcomes, making them an integral tool in mathematics and its real-world applications.