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Chapter 7: Problem 34

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -5 y=15 x+55 $$

Short Answer

Expert verified
Answer: The slope of the line is -3 and the y-intercept is -11.

Step by step solution

01

Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, we'll start by dividing both sides of the equation by -5. This will help to isolate y on the left side of the equation. $$ \frac{-5 y}{-5}=\frac{15 x + 55}{-5} $$ Now simplify the equation: $$ y= -3x -11 $$
02

Identify the slope and y-intercept

In the slope-intercept form, \(y = mx + b\), 'm' represents the slope and 'b' represents the y-intercept. Comparing this with our equation \(y = -3x - 11\), we can identify: $$ m = -3 \quad \text{and} \quad b = -11 $$ So, the slope of the line is -3 and the y-intercept is -11.

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Key Concepts

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

Understanding the Slope
When we talk about the slope of a line in a linear equation, we're referring to a number that describes the steepness or incline of the line. In the slope-intercept form of a linear equation, which is written as \( y = mx + b \), the letter \( m \) represents the slope. This value tells us how much \( y \) changes for a unit change in \( x \). For instance,
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • If the slope is zero, the line is perfectly horizontal.
In the given equation \( y = -3x - 11 \), the slope \( m \) is -3. This means for every 1 unit increase in \( x \), \( y \) decreases by 3 units. The negative sign indicates that the line slopes downward.
Identifying the Y-Intercept
The y-intercept in a linear equation is the point where the line crosses the y-axis. In slope-intercept form \( y = mx + b \), the \( b \) represents the y-intercept. This value is crucial because it shows the initial position of the line on a graph when \( x = 0 \). To find the y-intercept:
  • Set \( x \) to zero in the equation.
  • Solve for \( y \).
For example, in the equation \( y = -3x - 11 \), replacing \( x \) with 0 gives us \( y = -11 \). Thus, the y-intercept is \( -11 \). This means the line intersects the y-axis at the point (0, -11). It's a helpful way to graph the line quickly.
Basics of Linear Equations
Linear equations represent straight lines on a graph and are fundamental in algebra. The slope-intercept form, \( y = mx + b \), is a simple and effective way to express these equations:
  • It quickly provides both the slope and y-intercept, helping in graph plotting.
  • This form is highly intuitive for evaluating how changes in \( x \) affect \( y \).
Linear equations model everyday scenarios where there is a constant rate of change. For instance:
  • Calculating distance over time if speed is constant.
  • Determining cost per unit in budgeting scenarios.
By understanding these equations, you can solve practical problems and better understand mathematical relationships.

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Most popular questions from this chapter

Graph the equations. $$ y=0 $$

Find the slope, if it exists, of the line through the given pairs of points. $$ (1,-4), \quad(3,3) $$

Graph the inequalities. $$ x<3 $$

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ (2,3), \quad(3,5) $$

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=-11, \quad y\) -inter cept \(=-4\)
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