Problem 45 - Precalculus Essentials

Question

Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$y=-\frac{3}{5} x+7$$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{3}{5}\) and the y-intercept is \(7\).

Step 1: Identify the slope

From the equation \(y=-\frac{3}{5}x+7\), the slope (\(m\)) can be identified as the coefficient of \(x\), which in this case is \(-\frac{3}{5}\). So, the slope of the line is \(-\frac{3}{5}\).

Step 2: Identify the y-intercept

From the same equation, the y-intercept (\(b\)) can be identified as the constant, which in this case is \(7\). So, the line intercepts the y-axis at \(7\).

Step 3: Graph the function

Now we graph the line using the slope and the y-intercept. Start by marking a point on the y-axis at \(7\) (as this is the y-intercept). The slope is \(-\frac{3}{5}\), which means that for every 5 units we move to the right on the x-axis, we move down 3 units on the y-axis (because the slope is negative). So from our point on the y-intercept, move 5 units to the right and 3 units down, and mark another point. The line that passes through these two points represents the given equation.

Key Concepts

Understanding the Slope

The concept of 'slope' is fundamental to understanding how linear equations work. The slope in a linear equation is represented by the letter \( m \) when in the form \( y = mx + b \). It tells us how steep or flat a line is on a graph.

In our example with the equation \( y = -\frac{3}{5}x + 7 \), the slope \( m \) is \( -\frac{3}{5} \). This means every time we move 5 units to the right on the x-axis, we go down 3 units on the y-axis because the slope is negative. A positive slope would mean the line rises as it moves from left to right, whereas a negative slope means the line declines as it goes from left to right.

Key points to remember about slope:

• It tells us the direction and steepness of a line.
• A positive slope rises, while a negative slope falls as it moves rightward.
• A slope of zero means the line is horizontal, and an undefined slope means the line is vertical.

Decoding the Y-Intercept

The y-intercept is another crucial element in linear equations. It is the point where the line crosses the y-axis on a graph. In the equation, the y-intercept is represented by the symbol \( b \).

For the linear equation \( y = -\frac{3}{5}x + 7 \), the y-intercept is \( 7 \). This means the line will cross the y-axis at the point \( (0, 7) \). The y-intercept is straightforward to identify:

• It is the constant term in the equation form \( y = mx + b \).
• It shows what the value of y is when x equals 0.
• The point \( (0, b) \) always lies on the graph of the line.

This makes it easy to start graphing because you have a precise starting point.

Graphing Linear Functions Made Easy

Graphing a linear function means illustrating what the equation looks like on a graph. With the equation \( y = -\frac{3}{5}x + 7 \), we have both the slope and the y-intercept needed to create the graph.

Here's how to graph it step-by-step:

• Start by plotting the y-intercept. In this case, you will put a point at \( (0, 7) \) on the y-axis.
• Next, use the slope to plot additional points. With a slope of \( -\frac{3}{5} \), move 5 units to the right on the x-axis, then 3 units down on the y-axis from the y-intercept, and position another point.
• Draw a straight line through these points extending in both directions across the graph. This line represents the linear function.

These simple steps allow the accurate graphing of any linear equation, making visualization of mathematical relationships straightforward and intuitive. With these principles, you'll be graphing linear equations like a pro in no time!