Problem 25 - Calculus with Concepts in Calculus

Question

Determine the slope \(m\) and \(y\) intercept \(b\) of the line with the given equation. Then sketch the line. \(2 x+y-4=0\)

Step-by-Step Solution

Answer

The slope is -2, and the y-intercept is 4.

Step 1: Convert the Equation to Slope-Intercept Form

The equation given is in standard form: \(2x + y - 4 = 0\). The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To convert, solve for \(y\). Start by isolating \(y\) on one side:\[ y = -2x + 4 \].Now, the equation is in slope-intercept form.

Step 2: Identify the Slope and Y-Intercept

From the slope-intercept form of the equation \(y = -2x + 4\), we can directly read the slope \(m\) and the y-intercept \(b\). The slope \(m\) is the coefficient of \(x\), which is \(-2\), and the y-intercept \(b\) is the constant term, which is \(4\). So, \(m = -2\) and \(b = 4\).

Step 3: Sketch the Line

To sketch the line, start by plotting the y-intercept (0, 4) on the coordinate plane. Next, use the slope \(m = -2\), which is the ratio \(-2/1\). From the y-intercept, move down 2 units and right 1 unit to find the next point (1, 2). Connect these points with a straight line extending in both directions.

Key Concepts

Linear Equation

A linear equation is an algebraic equation that forms a straight line when graphed on a coordinate plane. These equations are typically written in the standard form, which is \[ ax + by + c = 0 \] and can be transformed into the slope-intercept form, such as \[ y = mx + b \]. Here, \(x\) and \(y\) are variables, while \(a\), \(b\), and \(c\) are constants.

• Linear equations represent relationships where, for every change in \(x\), there is a consistent change in \(y\).
• The standard form can be useful for identifying specific qualities of the line, like its slope and y-intercept when converted.

Linear equations are foundational in algebra as they model constant rates of change.

Slope

In mathematics, the slope of a line is a measure of its steepness and direction. From the slope-intercept form of a linear equation, \( y = mx + b \), the slope is represented by \(m\). It describes how much \(y\) changes for a change in \(x\).

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends from left to right, as in our example with a slope of \(-2\).
• A slope of zero indicates a horizontal line.

Calculating slope is fundamental for understanding how two variables relate within a linear format.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis of a coordinate plane. With the slope-intercept form \( y = mx + b \), the y-intercept is given by the constant \(b\). It indicates the value of \(y\) when \(x = 0\), essentially where the line starts on the y-axis.

In our exercise, the y-intercept is \(4\), meaning:

• The line crosses the y-axis at the point \((0, 4)\).
• This point is essential for sketching the graph of the line.

The y-intercept provides a starting point for visualizing a linear equation on a coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It allows us to graph equations and visualize relationships between variables. Each point on this plane is defined by an \((x, y)\) coordinate.

Key concepts to understand include:

• Quadrants: The plane is divided into four quadrants, which help to identify positive and negative values of \(x\) and \(y\).
• Origin: The point \((0,0)\) where the axes intersect.
• Axes: Serve as reference lines for graphing the points.

Understanding the coordinate plane is crucial for interpreting the geometric representation of equations.

Graphing Lines

Graphing lines involves plotting points on a coordinate plane, joining them in a straight path. Here's how you can approach it:

• Identify the y-intercept: Start at the point where the line crosses the y-axis.
• Use the slope: The slope indicates how to proceed from the y-intercept to find the next point on the line.

For example, with the equation \( y = -2x + 4 \),

• Begin at the y-intercept \((0, 4)\).
• Apply the slope \(-2\): Move down two units and one unit to the right to locate the next point, \((1, 2)\).

Connect these points with a line, extending in both directions, and you’ve successfully graphed your line. This visual representation helps in understanding the nature and implication of linear equations.