Problem 23 - Algebra 2

Question

Write an equation in slope-intercept form that satisfies each set of conditions. slope \(4,\) passes through \((0,6)\)

Step-by-Step Solution

Answer

The equation is \( y = 4x + 6 \).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula: \[ y = mx + b \]where \( m \) represents the slope and \( b \) represents the y-intercept.

Step 2: Insert the Given Slope

We know that the slope \( m \) is \( 4 \). Therefore, we substitute \( 4 \) for \( m \) in the equation:\[ y = 4x + b \]

Step 3: Determine the Y-intercept

The point \((0,6)\) means the line crosses the y-axis at \( y = 6 \). This tells us the y-intercept \( b \) is \( 6 \).

Step 4: Write the Final Equation

Substitute \( b = 6 \) back into the equation:\[ y = 4x + 6 \]This is the equation in slope-intercept form that passes through the point \((0,6)\) with a slope of \(4\).

Key Concepts

Linear Equation

A linear equation represents a straight line on a graph and can be expressed in various forms, one of the most common being the slope-intercept form. In mathematics, linear equations are written as expressions where each term is either a constant or the product of a constant and a single variable. These equations are significant because they help describe relationships with a constant rate of change. The general form of a linear equation is ⏦ax + by = c⏧, where ⏦a⏧, ⏦b⏧, and ⏦c⏧ are constants.
However, for easier analysis, especially in graphing, the slope-intercept form ⏦y = mx + b⏧ is often preferred. This form clearly indicates the slope and the y-intercept, making it straightforward to visualize the line it represents.

Slope

The slope of a line in geometry determines how steep the line is. It's often denoted by the letter ⏦m⏧ in the slope-intercept equation \(y = mx + b\).
The slope is calculated as the 'rise over run,' or the change in the ⏦y⏧-direction versus the change in the ⏦x⏧-direction between two points on a line.

• A positive slope means the line inclines upwards as you move from left to right.
• A negative slope indicates it declines.
• A slope of zero signifies that the line is horizontal, meaning no change in ⏦y⏧ as ⏦x⏧ changes.
• An undefined slope refers to a vertical line since there's no change in ⏦x⏧.

In the slope-intercept form equation given, ⏦m = 4⏧ indicates a line that rises 4 units upwards for every unit it moves to the right.

Y-intercept

The y-intercept of a linear equation is the point where the line crosses the ⏦y⏧-axis. In the equation \(y = mx + b \), the y-intercept is represented by the term ⏦b⏧.
This intercept provides a starting point on the graph for where the line will intersect the ⏦y⏧-axis, making it a crucial part of understanding the line's orientation on the graph.

• If ⏦b = 0⏧, the line passes through the origin.
• If ⏦b > 0⏧, the y-intercept will be above the origin.
• If ⏦b < 0⏧, it will be below the origin.

In the original problem, the point ⏦(0,6)⏧ directly tells us that the y-intercept ⏦b⏧ is 6. This provides critical information to fully define the linear equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry to study geometric shapes using a coordinate plane. This field of mathematics revolutionized how we visualize and solve geometric problems.
In a coordinate plane:

• The x-axis runs horizontally, while the y-axis runs vertically.
• Points are positioned using ordered pairs ⏦(x, y)⏧.
• Lines and curves are defined using equations.

For linear equations, coordinate geometry provides a clear method for placing, assessing, and solving lines on a grid. Our exercise uses coordinate geometry principles to determine and confirm the line passing through the point ⏦(0,6)⏧ with a slope of 4. This method helps visually comprehend the relationship between algebraic equations and geometric representations.