Problem 9 - Algebra 1

Question

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (-3,10), m=4 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (-3,10) and has a slope of 4, is \(y = 4x + 22\).

Step 1: Identify the given slope and point

The slope of the line, \(m\), is 4. The coordinates of the point \((x_1, y_1)\) through which the line passes are (-3,10).

Step 2: Substitute the values into the point-slope form

Given the point-slope form \(y - y_1 = m (x - x_1)\), replace \(m\) with 4, \(x_1\) with -3, and \(y_1\) with 10. This will give us the equation \(y - 10 = 4 (x - (-3))\).

Step 3: Simplify the equation

Simplify the equation by performing the operations. This gives us \(y - 10 = 4x + 12\).

Step 4: Final Equation

Finally, by rearranging the equation, the point-slope form of the line is \(y = 4x + 22\).

Key Concepts

Equation of a Line

The equation of a line is a mathematical description of a straight path, consisting of points that satisfy certain conditions. It ties together the slope and the coordinates of points on the line, offering a formula that can be used to determine any point along that line.
One of the most common representations of line equations is the point-slope form, given by:

• \( y - y_1 = m(x - x_1) \)

This form reveals essential details, such as the line's slope \( m \) and a specific point \( (x_1, y_1) \) on the line. This approach simplifies identifying any line if you know the slope and one point.
Understanding how the equation of a line relates the x and y coordinates is fundamental for solving and graphing linear problems.

Slope

The slope of a line is a measure of its steepness or incline. It is represented by the letter \( m \) in the equation of a line.
The slope determines the direction and angle of the line across the coordinate plane.
If you were to imagine the slope as a ratio, it's essentially how much the y-coordinate changes for a unit change in the x-coordinate. More mathematically, it's calculated as

• \( m = \frac{\Delta y}{\Delta x} \)

where \( \Delta y \) is the change in y-values, and \( \Delta x \) is the change in x-values.
This concept is crucial since different slopes imply different behaviors:

• A positive slope means the line goes upward as we move from left to right.
• A negative slope indicates the line goes downward.
• A zero slope produces a horizontal line.
• An undefined slope (division by zero) corresponds to a vertical line.

Coordinates

Coordinates are pairs of numbers that define the position of a point in a two-dimensional space, usually expressed as ordered pairs \( (x, y) \).
The first number \( x \) denotes the position relative to the horizontal axis, while the second number \( y \) indicates the vertical position.
These coordinates are essential when dealing with lines as they help specify particular points through which a line passes.
In the point-slope form equation, \( (x_1, y_1) \) denotes a known point on the line, playing a crucial role in forming the line's equation.
Understanding coordinates helps students visualize lines on the coordinate plane, enabling them to comprehend how lines interact with the graph's axes and other lines.

Linear Equations

Linear equations represent relationships with constant rates of change, forming straight lines when plotted on a graph.
A linear equation can appear in various forms:

• Point-slope form \( y - y_1 = m(x - x_1) \)
• Slope-intercept form \( y = mx + b \)
• Standard form \( Ax + By = C \)

Each form provides a different perspective on the line's characteristics. The point-slope form is particularly useful when you know a point on the line and its slope.
Linear equations are foundational in algebra due to their simplicity and broad application in modeling real-world phenomena where relationships remain consistent.
By understanding linear equations, students can solve problems involving direct relationships and make predictions based on linear data.