Problem 53 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. $$3 x+y-5=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -3x + 5\). The slope is \(-3\) and the y-intercept is \(5\).

Step 1: Rearrange the Equation Into Slope-Intercept Form

The given equation is \(3x + y - 5= 0\), which can be rearranged into the slope-intercept form \(y = mx + b\) by isolating \(y\). To do this, subtract \(3x\) from both sides of the equation to get \(y = -3x + 5\).

Step 2: Identify the Slope and Y-Intercept

From the equation \(y = -3x + 5\), the slope \(m\) is \(-3\) and the y-intercept \(b\) is \(5\). The slope shows the steepness and direction of the line, while the y-intercept is the point where the graph crosses the y-axis.

Step 3: Graph the Equation

To graph the equation, start by plotting the y-intercept at the point \(0,5\) on the y-axis. The slope is \(-3\), this tells us that for every unit we move to the right on the x-axis, we need to move 3 units down on the y-axis. Use these details to plot more points on the graph and draw the line.

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness or incline. It is often represented by the letter \( m \) in mathematical equations. In the example equation \( y = -3x + 5 \), the slope \( m \) is \(-3\). This numerical value of the slope indicates two important aspects of the line:

• Direction: A negative slope means the line slopes downwards from left to right.
• Rate of Change: The slope of \(-3\) indicates that for every one unit increase along the x-axis, the value of y decreases by three units.

The steeper the slope, the more extreme the incline. A slope of zero would suggest a perfectly horizontal line, while an undefined slope would occur with a vertical line.
Grasping the concept of slope helps in understanding how one variable changes with another, particularly in linear equations.

Exploring the Y-Intercept

The y-intercept is a crucial component in graphing linear equations, represented by the letter \( b \) in the equation form \( y = mx + b \). It denotes the point where the line crosses the y-axis. In the equation \( y = -3x + 5 \), the y-intercept is \(5\).

Let's visualize what this means:

• Location on the graph: A y-intercept of \(5\) simply states that this line will intersect the y-axis at the point \((0, 5)\).
• Starting Point: The y-intercept can be thought of as the starting point of a line when x is zero. It gives us the exact place the line begins on the y-axis.

Knowing the y-intercept allows you to quickly draw a point on the graph even before plotting any more points using the slope, providing a starting anchor point for your line.

Graphing Linear Equations

Graphing linear equations can be made simple with a clear understanding of both the slope and the y-intercept. When you have an equation in slope-intercept form \( y = mx + b \), like \( y = -3x + 5 \), these steps will help you plot the graph accurately:

• Start with the Y-Intercept: Plot the y-intercept on the graph. In this instance, begin by marking the point \((0, 5)\) on the y-axis.
• Apply the Slope: Use the slope \(-3\) to determine the direction of the line. From your initial point \((0, 5)\), since the slope is \(-3\), move 3 units downwards and 1 unit to the right to plot the next point.
• Connect the Dots: Draw a straight line through the points. This line represents the equation, and extends infinitely in both directions.

Understanding these steps gives a solid foundation in graphing linear equations, as it allows you to visualize the relationship between variables plotted on a graph.