Problem 50 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(1,\) passes through \((5,3)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = x - 2 \).

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identifying Known Values

From the problem, we know the slope \( m = 1 \) and the line passes through the point \( (5, 3) \). This means when \( x = 5 \), \( y = 3 \).

Step 3: Substituting Known Values

Substitute \( m = 1 \), \( x = 5 \), and \( y = 3 \) into the slope-intercept form to find \( b \). The equation becomes \( 3 = 1(5) + b \).

Step 4: Solving for the Y-Intercept

Solve the equation \( 3 = 5 + b \) to find \( b \). Subtract 5 from both sides to get \( b = 3 - 5 = -2 \).

Step 5: Writing the Final Equation

Now that we have the slope \( m = 1 \) and the y-intercept \( b = -2 \), we can write the equation of the line as \( y = x - 2 \).

Key Concepts

Equation of a Line

An equation of a line in coordinate geometry serves as a mathematical representation of a straight line. It allows us to understand how the position and slope of the line relate to the coordinate plane. The most commonly used form of the equation of a line is the slope-intercept form. This is expressed as:

• \( y = mx + b \)

Here, \( y \) and \( x \) are the coordinates of any point on the line, \( m \) represents the slope, and \( b \) is the y-intercept.The slope-intercept form is particularly useful because it directly provides you with the slope of the line and the point at which the line crosses the y-axis.By simply looking at an equation written in slope-intercept form, you can tell both the steepness of the line and its position relative to the y-axis.Using this form makes it easy for us to plot the line and clearly understand its behavior in the coordinate plane.

Slope

The slope of a line is a key concept in understanding linear equations. It determines how steep a line is and is often referred to as the "rise over run."In the slope-intercept form of a line, \( y = mx + b \), the variable \( m \) represents the slope. The slope tells us:

• How much the line rises (or falls) for each unit it moves to the right.
• A positive slope means the line is inclined upwards as you move along the x-axis.
• A negative slope means the line descends as you move along the x-axis.
• A slope of zero indicates a horizontal line.

Understanding the slope helps us quickly determine the direction and slant of the line, which is crucial in various applications such as physics, economics and other fields that use data modeling.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.When writing an equation of a line, determining the y-intercept involves plugging in the values of a known point (usually given in the problem) and solving for \( b \). Here's how to interpret \( b \):

• It's the value of \( y \) when \( x = 0 \).
• Represents the initial value or starting point of the line on the y-axis.

Identifying the y-intercept is crucial, as it allows you to plot the line accurately on a graph. It essentially provides you the point from which to start drawing the line, using the slope to see where it extends.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes a coordinate system to understand geometric figures such as lines and shapes. This approach bridges algebra and geometry, allowing us to visualize algebraic equations on a graph.Here's how it works with a line:

• The coordinate plane comprises two perpendicular lines—the x-axis (horizontal) and the y-axis (vertical).
• Any point on this plane is defined by a pair of coordinates \((x, y)\).
• With equations like \( y = mx + b \), you can easily plot a line by identifying the y-intercept and using the slope.

Coordinate geometry simplifies complex algebraic expressions by providing a visual component, making it easier to grasp relationships between equations and shapes.This technique is fundamental in various fields, including computer graphics, navigation, and architecture, where understanding spatial relationships is essential.