Problem 46 - Algebra and Trigonometry

Question

Find the slope and \(y\)-intercept of the line and draw its graph. \(-3 x-5 y+30=0\)

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{5}\) and the \(y\)-intercept is 6.

Step 1: Rearrange the Equation

First, rearrange the given equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Start by moving the terms involving \( x \) and the constant to the other side of the equation: \(-5y = 3x - 30 \).

Step 2: Solve for y

Divide all terms by \(-5\) to solve for \( y \). This gives you the equation in slope-intercept form: \( y = -\frac{3}{5}x + 6 \).

Step 3: Identify the Slope and y-Intercept

Now that you have the equation in the form \( y = mx + b \), identify the slope \( m \) and the \( y \)-intercept \( b \). Here, the slope \( m = -\frac{3}{5} \) and the \( y \)-intercept \( b = 6 \).

Step 4: Graph the Equation

To draw the graph, start by plotting the \( y \)-intercept \( (0, 6) \) on the y-axis. From this point, use the slope \( -\frac{3}{5} \): move down 3 units and to the right 5 units to find another point \( (5, 3) \). Draw a straight line through these points.

Key Concepts

Slope

The slope of a line is a crucial concept in algebra. It describes how steep a line is and its direction on a graph. The slope is often represented by the letter \( m \) in the slope-intercept form of a linear equation, which is \( y = mx + b \). For our specific equation, \( -3x - 5y + 30 = 0 \), we've rearranged this to \( y = -\frac{3}{5}x + 6 \). Here, the slope \( m \) is \(-\frac{3}{5}\).

• A negative slope, like \(-\frac{3}{5}\), means the line goes downwards from left to right.
• The absolute value of the slope tells us the steepness: the larger the absolute value, the steeper the line.

The slope \(-\frac{3}{5}\) indicates the line moves down 3 units vertically for every 5 units it moves to the right horizontally. This ratio is always constant for a straight line, showcasing a uniform rate of change.

Graph of a Linear Equation

The graph of a linear equation forms a straight line. It's one of the simplest forms of graphs because of its predictable linearity once you know the slope and the y-intercept.

• The slope \( m \) determines the line's angle or direction.
• The y-intercept \( b \) tells where the line crosses the y-axis.

With the slope-intercept equation \( y = -\frac{3}{5}x + 6 \), you can easily draft the graph. You start by plotting the \( y \)-intercept at \((0, 6)\). From there, apply the slope: go down 3 units, and move 5 units right to mark another point at \((5, 3)\). Once you have these two points, draw a line through them, and you have the graph of your linear equation. This straightforward process makes graphing linear equations efficient and clear.

Y-Intercept

The y-intercept is a pivotal part of the slope-intercept form \( y = mx + b \). It is represented by \( b \) and indicates the point where the graph intersects the y-axis. For the equation \( y = -\frac{3}{5}x + 6 \), the y-intercept \( b \) is 6.

• This means the line crosses the y-axis at the point \((0, 6)\).
• It's essential because it gives the starting point on the graph before the slope is applied.

When drafting the graph, the y-intercept is the first coordinate you plot on the y-axis. Understanding the y-intercept is critical as it allows you to anchor the line accurately on the graph. The rest of the line is defined by the slope emanating from this y-intercept point.