Problem 50 - Precalculus Mathematics for Calculus

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: Rewrite the Equation in Slope-Intercept Form

The given equation is \(-3x - 5y + 30 = 0\). To find the slope and \(y\)-intercept, we first rewrite it in the slope-intercept form, which is \(y = mx + b\). Start by isolating \(y\) on one side:Add \(3x\) to both sides:\[-5y = 3x - 30\]Finally, divide each term by \(-5\):\[y = -\frac{3}{5}x + 6\]

Step 2: Identify the Slope and y-Intercept

Now that the equation is in the form \(y = mx + b\), we can identify the slope \(m\) and \(y\)-intercept \(b\):- The slope \(m\) is \(-\frac{3}{5}\).- The \(y\)-intercept \(b\) is 6.

Step 3: Plot the y-Intercept on the Graph

To begin graphing, locate the \(y\)-intercept on the graph. The \(y\)-intercept is 6, so put a point at \((0, 6)\) on the \(y\)-axis.

Step 4: Use the Slope to Find Another Point

From the \(y\)-intercept, use the slope \(-\frac{3}{5}\) to find another point. The slope indicates a rise of \(-3\) and a run of \(5\), meaning you move down 3 units and right 5 units from the \(y\)-intercept. Starting from \((0, 6)\), move to \((5, 3)\). Plot this point.

Step 5: Draw the Line

Draw a straight line through the two points \((0, 6)\) and \((5, 3)\). This line represents the equation \(-3x - 5y + 30 = 0\).

Key Concepts

Slope

In a linear equation, the slope is a measure of how steep the line is. It tells us how the line rises or falls as it moves from left to right across the graph. Mathematically, the slope \( m \) is defined in the equation of the line \( y = mx + b \) where \( m \) represents the slope.

The slope is found by calculating the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run) between any two points on the line:

• A positive slope means the line goes up as it moves to the right.
• A negative slope indicates the line goes down as it moves to the right.
• If the slope is zero, the line is perfectly horizontal.
• If the slope is undefined, the line is vertical.

In our exercise, the slope of the equation \( y = -\frac{3}{5}x + 6 \) is \( -\frac{3}{5} \).

This means for every 5 units we move horizontally, the line drops 3 units vertically.

y-intercept

The y-intercept is a point where the line crosses the y-axis. It provides a starting point for graphing the entire line. In the linear equation \( y = mx + b \), the constant \( b \) is the y-intercept and it signifies the value of \( y \) when \( x = 0 \).

Here's why it’s useful:

• It gives a precise point \( (0, b) \) to plot on the graph without performing additional calculations.
• This point is crucial because it often forms one of the easiest points to locate and use when graphing a line.
• The y-intercept helps visualize where the line starts on the vertical axis, acting as a reference or anchor point for the line.

For our equation, \( y = -\frac{3}{5}x + 6 \), the y-intercept is 6. This tells us the line crosses the y-axis at the point \( (0, 6) \).

You begin plotting the line using this point.

Graphing Lines

Graphing lines involves turning the linear equation into a visual representation on a coordinate plane. This helps us see the relationship between variables and how they change with respect to each other.

Here's a simplified process:

• Start by plotting the y-intercept. For the equation \( y = -\frac{3}{5}x + 6 \), place a point at (0, 6).
• Next, use the slope to determine another point. The slope \( -\frac{3}{5} \) means from the point (0, 6), you move 3 units down and 5 units to the right, landing at (5, 3).
• Once you have both points plotted, draw a straight line through them. Extend this line across the graph to show the entire equation visually.

Graphing is helpful because:

• It provides a clear image of the linear equation and its characteristics.
• Allows you to see intersections with other lines or axes.
• Helps in predicting values for given x or y just by observing the line.

This straightforward approach aids in understanding the behavior and direction of linear equations in a visual form.