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Chapter 4: Problem 2

Give the slope and the \(y\)-intercept for each equation. a. \(y=-4-3 x\) b. \(2 x+7=y\) c. \(38 x-10 y=24\)

Short Answer

Expert verified
a. Slope: -3, y-intercept: -4; b. Slope: 2, y-intercept: 7; c. Slope: 3.8, y-intercept: -2.4.

Step by step solution

01

Identify the form of the equation

Each equation needs to be in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Rearrange equation (a)

Equation (a) is given as \( y = -4 - 3x \). Rewrite it as \( y = -3x - 4 \). This is already in slope-intercept form. The slope \( m \) is \(-3\) and the y-intercept \( b \) is \(-4\).
03

Rearrange equation (b)

Equation (b) is \( 2x + 7 = y \). Rearrange it to \( y = 2x + 7 \). It is in slope-intercept form where the slope \( m \) is \( 2 \) and the y-intercept \( b \) is \( 7 \).
04

Convert equation (c) to slope-intercept form

For equation (c), \( 38x - 10y = 24 \), we first need to solve for \( y \). Start by isolating \(-10y\) on one side: \( -10y = -38x + 24 \). Divide the entire equation by \(-10\) to solve for \( y \): \( y = \frac{38}{10}x - \frac{24}{10} \). Simplify to get \( y = 3.8x - 2.4 \). Hence, the slope \( m \) is \( 3.8 \) and the y-intercept \( b \) is \(-2.4\).

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Key Concepts

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

Understanding Slope
The slope of a line in the slope-intercept form equation, represented as \( y = mx + b \), is indicated by the letter \( m \). The slope tells us how steep the line is and in which direction it goes. It captures the concept of 'rise over run', essentially showing the change in \( y \) over the change in \( x \).
To visualize this, imagine climbing a hill:
  • If the slope is positive, it鈥檚 like walking uphill. For example, in equation (b) \( y = 2x + 7 \), the slope \( m \) is 2. This tells us the line goes up 2 units on the \( y \)-axis for every unit it moves right on the \( x \)-axis.
  • If the slope is negative, you are going downhill. Equation (a) \( y = -3x - 4 \) has a slope of \(-3\), indicating the line declines 3 units on \( y \) for each step right on \( x \).
  • A zero slope means a flat line, with no rise or fall.
Knowing the slope helps predict how the line behaves on a graph.
The Role of Y-Intercept
The \( y \)-intercept in the slope-intercept form \( y = mx + b \) is the constant \( b \). This point is crucial because it shows where the line crosses the \( y \)-axis. This value is important to immediately identify the starting point of the line on the graph.
In each case, this value is clear once the equation is in slope-intercept form:
  • For equation (a), \( y = -3x - 4 \), the \( y \)-intercept is \(-4\). This means the line touches the \( y \)-axis at the point (0, -4).
  • In equation (b), \( y = 2x + 7 \), the intercept is \( 7 \), crossing the \( y \)-axis at (0, 7).
  • Looking at equation (c), \( y = 3.8x - 2.4 \), sees the line hitting the \( y \)-axis at \(-2.4\).
Spotting the y-intercept quickly tells you about the position of the graph vertically, making it easier to sketch lines.
Rearranging Equations to Slope-Intercept Form
To find the slope and y-intercept, equations should ideally be in the \( y = mx + b \) form. This rearrangement makes the properties of the line clear and easy to identify right away.
Here鈥檚 how you do it:
  • Equation (a): The equation \( y = -4 - 3x \) looks differently at first but can be reshuffled as \( y = -3x - 4 \). Notice that you reorder terms to match the \( mx + b \) structure.
  • Equation (b): Given as \( 2x + 7 = y \), this already aligns with \( y = 2x + 7 \) where you swap sides to reveal the slope and intercept cleanly. Both sides equate already to the desired form.
  • Equation (c): Transforming \( 38x - 10y = 24 \) might seem challenging. Start by isolating \(-10y\), bringing \( x \) terms across, then divide through by \(-10\) to simplify: \( y = 3.8x - 2.4 \).
Rearranging equations takes practice but makes understanding line graphs much more straightforward.

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Most popular questions from this chapter

Write an equation in point-slope form for a line, given its slope and one point that it passes through. a. Slope 3 ; point \((2,5)\) b. Slope \(-5\); point \((1,-4)\)

The equation \(3 x+2 y=6\) is in standard form a. Find \(x\) when \(y\) is zero. Write your answer in the form \((x, y)\). What is the significance of this point? (a) b. Find \(y\) when \(x\) is zero. Write your answer in the form \((x, y)\). What is the significance of this point? (a) c. On graph paper, plot the points you found in \(10 \mathrm{a}\) and \(\mathrm{b}\) and draw the line through these points. (a) d. Find the slope of the line you drew in \(10 \mathrm{c}\) and write a linear equation in intercept form. e. On your calculator, graph the equation you wrote in \(10 \mathrm{~d}\). Compare this graph to the one you drew on paper. Is the intercept equation equivalent to the standard-form equation? Explain why or why not. f. Symbolically show that the equation \(3 x+2 y=6\) is equivalent to your equation from \(10 \mathrm{~d}\).

Find the equation of a line that a. Has a positive slope and a negative \(y\)-intercept. b. Has a negative slope and a \(y\)-intercept of 0 . c. Passes through the points \((1,7)\) and \((4,10)\). d. Passes through the points \((-2,10)\) and \((4,10)\).

\- APPLICATION The volume of a gas is \(3.50 \mathrm{~L}\) at \(280 \mathrm{~K}\). The volume of any gas is directly proportional to its temperature on the Kelvin scale \((\mathrm{K})\). a. Find the volume of this gas when the temperature is \(330 \mathrm{~K}\). b. Find the temperature when the volume is \(2.25 \mathrm{~L}\)

Show all steps for a symbolic solution to each problem. a. \(4+2.8 x=51\) b. \(38-0.35 x=27\) c. \(11+3(x-8)=41\) d. \(220-12.5(x-6)=470\)
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