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Chapter 7: Problem 20

Write an equation in slope-intercept form for each line. slope \(=3, y\) -intercept \(=5\)

Short Answer

Expert verified
The equation is \( y = 3x + 5 \).

Step by step solution

01

Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula: \ \[ y = mx + b \] \ where \( m \) is the slope and \( b \) is the y-intercept. In this exercise, we are given these values.
02

Substitute the Slope into the Equation

Replace \( m \) in the equation with the given value of the slope, which is 3. This gives us: \ \[ y = 3x + b \]
03

Substitute the Y-Intercept into the Equation

Replace \( b \) in the equation with the given y-intercept, which is 5. This gives us the final equation: \ \[ y = 3x + 5 \]
04

Verify the Equation

Ensure that the formulated equation matches the given conditions of having a slope of 3 and a y-intercept of 5. Indeed, the equation \( y = 3x + 5 \) satisfies both conditions.

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Key Concepts

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

Linear Equation
A linear equation is a fundamental concept in mathematics, describing a straight line on a graph. These equations are of the first degree, meaning they have no exponents higher than one. A basic linear equation can look like this: \( y = mx + b \).
  • Variables: The letters \( x \) and \( y \) are the variables that represent the coordinates on a Cartesian plane.
  • Solution: Solving a linear equation means finding the value of the variable that makes the equation true. Typically, one variable is expressed in terms of the other.
  • Graphing: When graphed, they produce a straight line, hence the name "linear."
Linear equations are used across different fields, including science and economics, to model relationships between two variables.
Understanding them is crucial for moving ahead in algebra and mathematics in general.
Slope
The slope of a line is a measure of its steepness and is represented by the letter \( m \) in the slope-intercept form of a linear equation. It indicates how much the y-value changes for a change in the x-value.
  • Rise over Run: The slope is calculated as "rise over run," meaning the change in y divided by the change in x, or \( m = \frac{\Delta y}{\Delta x} \).
  • Types of Slopes: A positive slope indicates that the line ascends from left to right, while a negative slope means it descends.
  • Horizontal and Vertical Lines: A slope of zero means the line is horizontal, and an undefined slope corresponds to a vertical line.
The slope is key for understanding the direction and angle of the line. It allows us to compare how different lines relate to each other in terms of steepness.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis on a graph, signified by the variable \( b \) in the equation \( y = mx + b \). It represents the value of \( y \) when \( x = 0 \).
  • Point Definition: The y-intercept is expressed as the coordinates \( (0, b) \).
  • Interpretation: Understanding the y-intercept helps interpret the starting point of a linear relationship in real-world scenarios.
  • Graphical Representation: On a graph, look for the spot where the line meets the y-axis.
The y-intercept is crucial when you need to set initial conditions or starting values in various contexts, such as when predicting future events or analyzing past data in studies.

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Most popular questions from this chapter

Graph each equation using the slope and \(y\) -intercept. $$x+y=-3$$

Write an equation in slope-intercept form for the line passing through each pair of points. $$(5,-6) \text { and }(3,2)$$

A CD player has a pre-sale price of \(\$ c\). Kim buys it at a \(30 \%\) discount and pays \(6 \%\) sales tax. After a few months, she sells it for \(\$ d\) which was \(50 \%\) of what she paid originally. Express \(d\) as a function of \(c\).

Monica has 440 dollars to pay a painter to paint her bedroom. The painter charges 55 dollars per hour. The equation \(y=440-55 x\) represents the amount of money left after \(x\) number of hours worked by the painter. What does the solution \((7,55)\) represent? \(\mathbf{F}\) Monica has 7 dollars left after 55 hours of painting. G Monica has 55 dollars left after 7 hours of painting. H The job is completed after 7 hours. J The job is completed after 55 hours.

Subtract. $$15-31$$
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