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Chapter 1: Problem 31

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(\frac{2 x}{3}-y=1\)

Short Answer

Expert verified
Slope \( m = \frac{2}{3} \), y-intercept \( b = -1 \), and graph the line.

Step by step solution

01

Convert to Slope-Intercept Form

The given equation is \( \frac{2x}{3} - y = 1 \). To find the slope and y-intercept, we need to convert it to the slope-intercept form \( y = mx + b \). Begin by isolating \( y \) on one side of the equation. Add \( y \) and subtract 1 from both sides: \( \frac{2x}{3} - 1 = y \). Rearranging, we get \( y = \frac{2x}{3} - 1 \).
02

Identify the Slope and Y-intercept

From the equation \( y = \frac{2x}{3} - 1 \), we can directly identify the slope \( m \) and the y-intercept \( b \). The slope \( m \) is \( \frac{2}{3} \) and the y-intercept \( b \) is \( -1 \). Thus, the y-intercept point is \((0, -1)\).
03

Plot the Y-intercept on the Graph

On a graph, locate the y-intercept \((0, -1)\). This is where the line will cross the y-axis. Mark this point clearly on the graph.
04

Use the Slope to Find Another Point

To use the slope \( \frac{2}{3} \), start from the y-intercept \((0, -1)\). The slope tells us that for every 3 units moved to the right (positive x-direction), the line moves up 2 units (positive y-direction). So, from \((0, -1)\), move 3 units right to \((3, -1)\) and then 2 units up to \((3, 1)\). Mark the point \((3, 1)\) on the graph.
05

Draw the Line

With the two points \((0, -1)\) and \((3, 1)\) plotted, use a ruler to draw a straight line through these points. This line is the graph of the equation \( \frac{2x}{3} - y = 1 \). Make sure to extend the line and draw arrows at both ends to indicate that it continues infinitely in both directions.

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way to express a linear equation. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis. Converting an equation into this form helps us easily identify these two important characteristics. To transform an equation into slope-intercept form, we aim to isolate \( y \) by performing algebraic operations on the equation.

For example, when given an equation like \( \frac{2x}{3} - y = 1 \), adding \( y \) to both sides and subtracting 1 yields \( \frac{2x}{3} - 1 = y \), or rearranged as \( y = \frac{2x}{3} - 1 \). Now, the equation is clearly in slope-intercept form.
graphing lines
Graphing lines is about plotting a straight path on the coordinate plane using points derived from a linear equation. When you have an equation in slope-intercept form \( y = mx + b \), the task becomes more straightforward.

Begin by plotting the y-intercept, \((0, b)\), on the y-axis. This point is where your graph will intersect the y-axis. Once you have this point, use the slope \( m \), which is the rise over run formula, to find additional points. By continuously applying the slope rule鈥攎oving up by the rise value and right by the run value鈥攜ou can plot several points for accuracy.
  • Plot point unraveling from the y-intercept using the slope.
  • Draw a line through the points extending it beyond them.
  • Add arrows on both sides to show the line is infinite.
Graphing helps in visualizing the linear relationship represented by the equation.
finding the slope
Finding the slope of a line is critical as it shows the line's steepness and direction. In the equation \( y = mx + b \), \( m \) is the slope. It tells us how much \( y \) changes for a change in \( x \). Slope is calculated as \( \frac{rise}{run} \), indicating how far up (or down) we move for a unit horizontally.

For example, with the slope \( \frac{2}{3} \):
  • You move 2 units up for every 3 units right.
  • If the slope was negative, like \( -\frac{2}{3} \), you'd move 2 units down for every 3 units right.
This concept is vital in understanding line behavior and is especially useful when graphing.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis, represented as \( (0, b) \). In slope-intercept form \( y = mx + b \), \( b \) is the y-intercept.

Finding the y-intercept involves substituting \( x = 0 \) in the equation since this substitution ensures the line meets the y-axis. For the equation \( y = \frac{2x}{3} - 1 \), the y-intercept \( b \) is \( -1 \). Therefore, the intercept is at the point \((0, -1)\).
  • Identify \( b \) from the equation.
  • Plot the point on the graph.
  • It acts as a starting point for graphing the equation using the slope.
Understanding the y-intercept helps in graph construction and offers insights into the line's position in relation to the origin.

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Most popular questions from this chapter

BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)

Fill in the missing words: If a line slants downward as you go to the right, then its_______ is________.

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=3 x^{2}-5 x+2 $$

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{3}{x} $$

Use your graphing calculator to graph the following four equations simultaneously on the window \([-10,10]\) by \([-10,10]:\) $$ \begin{array}{l} y_{1}=3 x+4 \\ y_{2}=1 x+4 \end{array} $$ \(y_{3}=-1 x+4\) (Use \((-)\) to get \(-1 x\). \()\) $$ y_{4}=-3 x+4 $$ a. What do the lines have in common and how do they differ? b. Write the equation of a line through this \(y\) -intercept with slope \(\frac{1}{2}\). Then check your answer by graphing it with the others.
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