Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 41-44: Use the slope-intercept form to find the slope and \(y\)-intercept of the given line, and sketch its graph. $$ 7 x=-4 y-8 $$

Step-by-Step Solution

Answer

Slope: \(-\frac{7}{4}\); Y-intercept: \(-2\).

Step 1: Convert to Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Start by rearranging the given equation \( 7x = -4y - 8 \) into this form. We need to solve for \( y \). First, add \( 4y \) to both sides to isolate terms with \( y \) on one side:\[ 7x + 8 = -4y \]Next, divide each term by \(-4\) to solve for \( y \):\[ y = -\frac{7}{4}x - 2 \]

Step 2: Identify the Slope and Y-intercept

In the slope-intercept form \( y = mx + b \), the slope \( m \) is the coefficient of \( x \) and the \( y \)-intercept \( b \) is the constant term. From the equation \( y = -\frac{7}{4}x - 2 \), the slope \( m \) is \(-\frac{7}{4}\) and the \( y \)-intercept \( b \) is \(-2\).

Step 3: Sketch the Graph

To sketch the graph of the line, start by plotting the \( y \)-intercept on the graph. The \( y \)-intercept is \(-2\), so mark the point \((0, -2)\) on the \( y \)-axis. Next, use the slope \( m = -\frac{7}{4} \), which means for every 4 units you move to the right (positive \( x \)-direction), you move 7 units down (negative \( y \)-direction). From the point \((0, -2)\), move 4 units right and 7 units down to find the next point on the line. Connect these two points to draw the line.

Key Concepts

Slope Calculation

Understanding how to calculate the slope from an equation is a crucial skill in algebra. The slope indicates the steepness or incline of a line and is a measure of how much the line rises or falls as you move from left to right. It's often denoted by the letter \( m \).

To find the slope from an equation in the general form, like \( 7x = -4y - 8 \), we first need to rearrange it into the slope-intercept form, \( y = mx + b \). This requires isolating \( y \) on one side of the equation:

• Add \( 4y \) to both sides: \( 7x + 8 = -4y \).
• Divide each term by \(-4\) to solve for \( y \): \( y = -\frac{7}{4}x - 2 \).

Here, the coefficient of \( x \), \(-\frac{7}{4}\), becomes the slope \( m \). This tells us that for every 4 units the line moves horizontally (along the \( x \)-axis), it moves 7 units vertically (along the \( y \)-axis). A negative slope indicates that the line falls as it moves from left to right.

Y-intercept Identification

Identifying the \( y \)-intercept in an equation is straightforward once the equation is in slope-intercept form, \( y = mx + b \). The \( y \)-intercept is represented by \( b \), which is the value of \( y \) when \( x \) is zero. It gives us the point where the line crosses the \( y \)-axis.

From the equation \( y = -\frac{7}{4}x - 2 \), we identify that \( b = -2 \). This means the line intersects the \( y \)-axis at the point \( (0, -2) \).

Understanding this concept is essential because the \( y \)-intercept provides a starting point when you're sketching graphs, and it's a key component in understanding shifts in line positions. Think of it as your anchor point on the graph. In real-world applications, the \( y \)-intercept can represent an initial amount or condition, serving as a baseline before any changes occur.

Graph Sketching

Sketching the graph of a line using the slope-intercept form is a visual way of understanding linear equations. Once you've determined the slope and \( y \)-intercept, you can easily draw your line.

Follow these steps to sketch the graph of the given line \( y = -\frac{7}{4}x - 2 \):

• Start at the \( y \)-intercept \( (0, -2) \). Plot this point on the graph.
• Use the slope \( -\frac{7}{4} \). This informs you to move 4 units to the right along the \( x \)-axis and then 7 units down along the \( y \)-axis.
  
  This movement results from the slope's "rise over run" interpretation, where the negative sign indicates a downward movement.
• Plot your second point after moving 4 units right and 7 units down from \( (0, -2) \).
• Draw a straight line connecting these two points, extending it across the graph to determine the line's entire path.

By following these steps, you sketch the line and get a clearer visual interpretation of the algebraic equation. This technique is invaluable when checking solutions or visualizing data trends.