Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. (0,3) with a slope of \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{2}{3}x + 3 \).

Step 1: Understand the point-slope formula

The point-slope formula is used to find the equation of a line when given a point and the slope. It is written as \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope, and \((x_1, y_1)\) represents the coordinates of the given point.

Step 2: Identify the given values

We are given a slope \( m = \frac{2}{3} \) and a point \((x_1, y_1) = (0, 3)\). We will use these values to plug into the point-slope formula.

Step 3: Substitute into the point-slope formula

Substitute the given point and slope into the point-slope formula: \( y - 3 = \frac{2}{3}(x - 0) \).

Step 4: Simplify the equation

Distribute the slope on the right side: \( y - 3 = \frac{2}{3}x \).

Step 5: Solve for y to express in slope-intercept form

Add 3 to both sides of the equation to isolate \( y \): \( y = \frac{2}{3}x + 3 \). This is the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental way to express linear equations, which makes it very easy and intuitive to graph them. This form of a linear equation is written as:

\[ y = mx + b \]
Here, \( y \) is the dependent variable (often represented on the y-axis), and \( x \) is the independent variable (along the x-axis). The letters \( m \) and \( b \) have special meanings:

• \( m \) is the slope of the line, which indicates its steepness or incline.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

The slope \( m \) tells how much \( y \) changes for a change in \( x \); a larger value means a steeper incline. The y-intercept \( b \) represents the starting point of the line on the vertical axis when \( x = 0 \).
Converting from point-slope to slope-intercept form involves solving the equation to make \( y \) the subject.

Linear Equations

Linear equations are algebraic expressions representing straight-line relationships between two variables. These equations produce straight lines when graphed on a coordinate plane. A typical linear equation in two variables like \( x \) and \( y \) is:

\[ ax + by = c \]
Where \( a \), \( b \), and \( c \) are constants. The slope-intercept form \( y = mx + b \) is a streamlined version specifically designed for ease of graphing and interpretation.
In these equations:

• \( a \) and \( b \) denote coefficients that affect how the line looks.
• \( c \) relates to the y-intercept when adjusting the equation.

The key characteristic of linear equations is their constant rate of change. They plot as straight lines, and any point on the line will satisfy the equation, making them predictable and easy to work with in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometrical figures using a coordinate system. It merges algebra and geometry to analyze the properties of figures on the plane.
The coordinate plane consists of two perpendicular axes:

• The horizontal axis (x-axis).
• The vertical axis (y-axis).

Each point on the plane is described by coordinates \((x, y)\), which specify its position relative to the axes. This framework allows us to explore geometric concepts algebraically.
In the context of linear equations:

• The coordinates form points on a line dictated by the equation.
• The slope represents the angle and direction of the line.

Coordinate geometry makes it possible to graph lines, find distances between points, and determine midpoints, linking algebraic expressions to visual interpretations on the plane.