Problem 4 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(2,-4), m=-5$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -5x + 6\).

Step 1: Insert the provided values into the slope-intercept formula.

Firstly, substitute the given values of the point and the slope into the slope-intercept equation \(y = mx + b\). This results in: \(-4 = -5*(2) + b\).

Step 2: Simplify the equation in order to determine the y-intercept.

Next, simplify this equation. Multiply -5 (the slope) by 2 (the x-coordinate) to get -10. This results in: \(-4 = -10 + b\). Then, add 10 to both sides of the equation to solve for b: \(b=-4+10\).

Step 3: Solve for the y-intercept.

Solve the equation for b. This results in: \(b = 6\).

Step 4: Construct the final equation.

Now that the slope (m) and the y-intercept (b) are known values, they can be inserted into the slope-intercept equation. Therefore, the final equation of the line in slope-intercept form is: \(y = -5x + 6\).

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that describe straight lines on a coordinate plane. These equations are typically written in one of several forms, with the most common being the slope-intercept form, which is written as \( y = mx + b \). In this formula, \( y \) represents the dependent variable or the vertical axis, \( x \) is the independent variable, or the horizontal axis.
This form is incredibly useful because it directly reveals two key components of a line: the slope \( m \) and the y-intercept \( b \). Linear equations are called “linear” because the graph of these equations forms a straight line. Understanding how to manipulate and interpret these equations is valuable for many applications, such as predicting trends and modeling real-world scenarios.

• Linear equations are used to describe relationships between two quantities, representing them graphically as lines.
• The slope-intercept form is particularly favored for its simplicity and ease of use in identifying the slope and y-intercept quickly.

Mastering linear equations lays a solid foundation for further study in mathematics, including functions and calculus.

Slope

The slope of a line is a measure of its steepness and direction, given by the variable \( m \) in the equation \( y = mx + b \). It is calculated as the ratio of the change in the vertical direction to the change in the horizontal direction between two points on the line. Mathematically, it is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), meaning the change in \( y \) over the change in \( x \).
Slope can tell us if a line is rising, falling, horizontal, or vertical on a graph:

• If \( m > 0 \), the line rises from left to right, indicating a positive slope.
• If \( m < 0 \), the line falls from left to right, indicating a negative slope.
• If \( m = 0 \), the line is horizontal, meaning no rise or fall as you move along the x-axis.
• Vertical lines have an undefined slope, as the change in x would be zero, leading to division by zero.

Understanding slope is essential for graphing lines and analyzing the relationships between variables. It helps predict patterns and can even provide insights into the speed or rate at which changes occur in various contexts.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by the variable \( b \). This value shows where the line meets the vertical axis when the value of \( x \) is zero.
The y-intercept is crucial because it provides a concrete starting point for graphing a line. Simply plot the y-intercept on the y-axis, and then use the slope to find other points on the line, moving up or down and left or right according to its steepness.

• A y-intercept of zero implies the line passes through the origin \((0,0)\) on the graph.
• The y-intercept helps in quickly sketching and understanding the behavior of a line within a graph.

The concept of the y-intercept is foundational in linear algebra and useful in various fields, from economics to physics, wherever linear models are applied to interpret data trends and relationships.