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Chapter 2: Problem 42

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

Short Answer

Expert verified
Slope is \(-\frac{7}{4}\) and the y-intercept is -2.

Step by step solution

01

Rearrange the Equation

To find the slope and the y-intercept, we need to write the given equation in the slope-intercept form, which is \(y = mx + b\). Start by rearranging the given equation \(7x = -4y - 8\). First, isolate \(y\) by adding \(4y\) to both sides to get \(7x + 4y = -8\). Next, we need to solve for \(y\).
02

Solve for y

Now solve the equation \(7x + 4y = -8\) for \(y\). Subtract \(7x\) from both sides to get \(4y = -7x - 8\). Then, divide every term by 4 to solve for \(y\): \(y = -\frac{7}{4}x - 2\). Now the equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
03

Identify Slope and Y-intercept

From the slope-intercept form \(y = -\frac{7}{4}x - 2\), identify the slope \(m\) as \(-\frac{7}{4}\) and the y-intercept \(b\) as \(-2\) (0, -2). This tells us the line decreases since the slope is negative and passes through the y-axis at -2.
04

Sketch the Graph

To sketch the graph, begin at the y-intercept of -2 on the y-axis. From this point, use the slope \(-\frac{7}{4}\). This means for a run (rightward movement) of 4 units, the line will fall (downward movement) 7 units. Plot the second point accordingly and draw a line through these two points, extending in both directions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Linear Equations
Graphing linear equations involves plotting points on a graph that represents solutions of the equation. The primary goal is to create a straight line that visually represents all possible solutions. To begin, it's crucial to transform the equation into a more graphable form, typically the slope-intercept form:
  • This form is expressed as \( y = mx + b \).
  • The variable \( y \) represents the dependent variable, while \( x \) is the independent variable.
  • Understanding this format allows us to easily identify the slope and y-intercept, making graphing straightforward.
Breaking down a problem often involves rearranging the given equation to this form, allowing one to sketch its graph efficiently. For example, the equation \( 7x = -4y - 8 \) can be rearranged to \( y = -\frac{7}{4}x - 2 \) to identify its linear characteristics.
Slope of a Line
The slope of a line, represented by \( m \) in the equation \( y = mx + b \), indicates the steepness and direction of the line. It reveals how much the dependent variable \( y \) changes for a unit change in \( x \). In our example equation, the slope is \(-\frac{7}{4}\). Here's what this means:
  • A slope of \(-\frac{7}{4}\) tells us that for every 4 units moved horizontally (in the x-direction), the y-value decreases by 7 units.
  • The negative sign indicates a downward trend as you move from left to right on the graph.
  • The greater the magnitude of the slope, the steeper the line. Conversely, a slope near zero would make the line more horizontal.
Understanding the slope provides insight into the nature of the relationship between variables in the equation, illustrating whether they are positively or negatively associated.
Y-intercept
The y-intercept is the point where the line crosses the y-axis. This gives the value of y when \( x \) is zero and is represented by \( b \) in the slope-intercept form.
  • In the given equation \( y = -\frac{7}{4}x - 2 \), the y-intercept is \(-2\).
  • This implies that the line will pass through the point (0, -2) on the graph.
  • The y-intercept is a crucial starting point for plotting the line, providing a fixed point from which to apply the slope.
Identifying this point allows one to anchor the line on the graph before using the slope to determine the line's direction and extend it across the graph.
Linear Equations
Linear equations form the foundation for understanding straight lines on a graph. These equations establish a linear relationship between variables and are generally characterized by their first-degree format. Such equations are vital in many fields, including physics, economics, and mathematics.
  • They are defined by having a constant rate of change, which is reflected in the slope.
  • The general form of linear equations is \( Ax + By = C \) (standard form) or \( y = mx + b \) (slope-intercept form).
  • A linear equation typically results in a straight line when graphed, showcasing a consistent, predictable relationship between variables.
Recognizing and manipulating these equations enables individuals to model real-world scenarios, make predictions, and analyze data effectively. The transformation of equations into different forms, as seen with \( 7x = -4y - 8 \), showcases the equation's linear attributes, simplifying both interpretation and graphing tasks.

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Most popular questions from this chapter

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