Problem 47 - Calculus Single Variable

Question

Sketch the line whose Cartesian equation is given. $$ 2 x+3 y=6 $$

Step-by-Step Solution

Answer

The line crosses the y-axis at (0, 2) and the x-axis at (3, 0).

Step 1: Identify the Equation Type and Components

The given equation is a linear equation of the form \( 2x + 3y = 6 \). Here, \( 2x \) and \( 3y \) are the linear components, and the equation is in standard form \( Ax + By = C \), where \( A = 2 \), \( B = 3 \), and \( C = 6 \).

Step 2: Rewrite the Equation in Slope-Intercept Form

To find the slope and y-intercept, rewrite the equation in the form \( y = mx + b \). Solve for \( y \):\[2x + 3y = 6\]Subtract \( 2x \) from both sides:\[3y = -2x + 6\]Divide each term by 3:\[y = -\frac{2}{3}x + 2\]

Step 3: Identify the Slope and Y-Intercept

In the slope-intercept form \( y = mx + b \), the slope \( m \) is \(-\frac{2}{3}\) and the y-intercept \( b \) is 2. This means that the line crosses the y-axis at (0, 2) and has a slope of \(-\frac{2}{3}\), indicating it decreases by 2 units for every 3 units it moves to the right.

Step 4: Find the X-Intercept

To find the x-intercept, set \( y = 0 \) in the original equation and solve for \( x \):\[2x + 3(0) = 6\]\[2x = 6\]\[x = 3\]So, the line crosses the x-axis at (3, 0).

Step 5: Sketch the Line

Use the intercepts found to draw the line. Mark the y-intercept (0, 2) and the x-intercept (3, 0) on a Cartesian plane. Draw a straight line through these points; this represents the graph of \( 2x + 3y = 6 \).

Key Concepts

Cartesian Plane

Think of the Cartesian Plane as a two-dimensional field where any point can be described using an ordered pair of numbers, known as coordinates. This plane consists of two main axes: the horizontal axis, called the x-axis, and the vertical axis, called the y-axis. The point where these axes intersect is known as the origin, labeled as (0, 0). Each point on the plane is given by an x-coordinate and a y-coordinate, which respectively tell you how far to move from the origin along the x-axis and the y-axis.

:

• The x-coordinate shows the horizontal distance from the origin.
• The y-coordinate shows the vertical distance from the origin.
• The intersection of the axes is the origin, (0, 0).

To graph an equation on the Cartesian Plane, we plot points that satisfy the equation. Once enough points are plotted, a straight line or a curve can be drawn to represent the equation. In this exercise, knowing coordinates helps us locate intercepts accurately when sketching graphs.

Slope-Intercept Form

The Slope-Intercept Form is a way to express the equation of a line, specifically aimed at revealing two key characteristics: the slope and the y-intercept.

Given in the format \[ y = mx + b \] where:

• \( m \) represents the "slope" of the line. The slope shows how steep the line is and determines the direction the line is slanting.
• \( b \) is the "y-intercept," showing the point where the line crosses the y-axis.

The slope indicates how much the y-value changes for a given change in the x-value. In simpler terms, if you move one unit horizontally, "\( m \)" tells you how much you'll go up or down. The y-intercept, \( b \), is the y-coordinate of the location where the line meets the y-axis. In our exercise's example, when rewritten as \[ y = -\frac{2}{3}x + 2 \], the slope is -\( \frac{2}{3} \), and the y-intercept is 2.

Intercepts

Intercepts are where a line crosses the axes on the Cartesian Plane. They help in quickly sketching lines because these points are typically easy to identify.

Y-intercept:

• This is where the line crosses the y-axis. It has the general formula (0, \( b \)), where \( b \) is the constant in the slope-intercept form.

In the exercise, the y-intercept (0, 2) shows that the line passes through 2 on the y-axis.

X-intercept:

• This is where the line crosses the x-axis. You find it by setting \( y = 0 \) in the equation and solving for \( x \).

For our example, the x-intercept at (3, 0) indicates the line meets the x-axis at 3. Intercepts are particularly useful as starting points when graphing a line, allowing you to pinpoint where to begin your drawing.

Graphing

Graphing is the process of visually representing the equation of a line on the Cartesian Plane using points and the line that connects them. By graphing, you transform an equation into a visual picture, making it easier to analyze and understand the relationship between x and y.

Using the information from the slope-intercept form and the intercepts, graphing becomes much simpler. Here's how you proceed:

• Start by plotting the intercepts: First, identify the y-intercept and mark it on the y-axis. Then, find the x-intercept and place a point on the x-axis.
• Draw the line: Use a ruler to connect these points with a straight line. This visual appreciation of the line shows the equation's solutions, where any point on this line satisfies the equation.
• Slope tells direction: The slope indicates whether the line ascends or descends as you move from left to right.

Graphing brings together all the concepts, offering a comprehensive view of how x and y relate through the linear equation, like in the given exercise for creating a precise line.