Problem 50 - 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

Slope: \(-\frac{3}{5}\); Y-intercept: 6; Graph by plotting (0,6) and using slope.

Step 1: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by manipulating the given equation to match this form. The given equation is \(-3x - 5y + 30 = 0\). To isolate \( y \), first move \(-3x\) and \(30\) to the other side by adding them: \(-5y = 3x - 30\).

Step 2: Solve for y

Continue solving for \( y \) by dividing every term by \(-5\) to get \( y = -\frac{3}{5}x + 6\). This is the slope-intercept form of the line where the coefficient of \( x \) represents the slope \( m \) and the constant term is the y-intercept \( b \).

Step 3: Identify Slope and Y-Intercept

From the slope-intercept form \( y = -\frac{3}{5}x + 6 \), identify that the slope \( m \) is \( -\frac{3}{5} \) and the y-intercept \( b \) is \( 6 \).

Step 4: Graph the Line

Begin by plotting the y-intercept \( (0, 6) \) on the graph. Use the slope \( -\frac{3}{5} \), which means "down 3 units, right 5 units," to find another point. From \( (0, 6) \), move down 3 units and 5 units to the right to reach \( (5, 3) \). Draw a straight line through these points to graph the line.

Key Concepts

Y-Intercept

A line's y-intercept is where it crosses the y-axis. This point is pivotal in graphing linear equations, as it provides a starting point to plot the line. In slope-intercept form, expressed as \( y = mx + b \), the \( b \) value is the y-intercept.
To understand further, consider that when \( x = 0 \), the equation simplifies to \( y = b \). This means the line intercepts the y-axis at \( (0, b) \).
In our example equation \( y = -\frac{3}{5}x + 6 \), \( b \) is \( 6 \). Consequently, the y-intercept is at the point \( (0, 6) \).
This point is always plotted first when graphing, helping guide the graph's orientation and position.

Slope

The slope of a line indicates its steepness and direction. Represented by \( m \) in the slope-intercept form \( y = mx + b \), slope reflects how much the y-value changes for a given change in the x-value.
For instance, a slope of \( -\frac{3}{5} \) signifies that for every 5 units you move to the right along the x-axis, you move 3 units down on the y-axis. A positive slope means the line ascends, while a negative slope implies it descends.

• Zero slope: Line is horizontal.
• Undefined slope: Line is vertical.

In our example, the line's slope is \( -\frac{3}{5} \). This negative value tells us the line moves downward as it goes from left to right.

Graphing Linear Equations

Graphing linear equations involves plotting a line on a coordinate plane that reflects the equation's slope and y-intercept.
This process starts with transforming the equation into the slope-intercept form \( y = mx + b \). For our line, the equation is \( y = -\frac{3}{5}x + 6 \).
Steps to graphing:

• First, plot the y-intercept. Here, the intercept is \( (0, 6) \).
• Use the slope to find another point. For \( -\frac{3}{5} \), start at \( (0, 6) \), move 3 units down and 5 units right to arrive at \( (5, 3) \).
• Draw a straight line through these points.

This method effectively translates an equation into a visual representation, making it easier to analyze and understand mathematical relationships.