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Chapter 1: Problem 28

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

Short Answer

Expert verified
Slope is \(\frac{2}{3}\) and \(y\)-intercept is \((0, -2)\).

Step by step solution

01

Simplify the equation

The given equation is \(y = \frac{2}{3}(x - 3)\). First, distribute the \(\frac{2}{3}\) to both \(x\) and \(-3\) inside the parentheses:\[ y = \frac{2}{3}x - \frac{2}{3} \times 3. \] This simplifies to \[ y = \frac{2}{3}x - 2. \] Now, the equation is in the slope-intercept form \(y = mx + b\).
02

Identify the slope and y-intercept

In the equation \(y = \frac{2}{3}x - 2\), compare it to the standard slope-intercept form \(y = mx + b\). Here the slope \(m\) is \(\frac{2}{3}\) and the \(y\)-intercept \(b\) is \(-2\). The \(y\)-intercept as a point is \((0, -2)\).
03

Plot the y-intercept

On a coordinate plane, mark the \(y\)-intercept \((0, -2)\). This is the point where the line crosses the \(y\)-axis.
04

Use the slope to find another point

The slope \(\frac{2}{3}\) means rise over run. From the \(y\)-intercept \((0, -2)\), move up 2 units (rise) and 3 units to the right (run) to find another point, \((3, 0)\).
05

Draw the line

Plot the points \((0, -2)\) and \((3, 0)\) on the graph. Connect these points with a straight line to represent the equation \(y = \frac{2}{3}x - 2\). The line represents all solutions to the equation.

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions that represent straight lines on a coordinate plane. They take the general form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is particularly useful as it directly gives us important information needed to graph these equations.
  • The slope \( m \) indicates how steep the line is and in which direction it moves.
  • The y-intercept \( b \) tells us where the line crosses the y-axis.
The simplicity of the slope-intercept form allows for quick identification of these components, making it easier to graph the equation and understand the relationship between variables.
Graphing
Graphing linear equations involves plotting points on a coordinate plane to visually represent the equation. To graph, we need:
  • The slope \( m \).
  • The y-intercept \( b \).
Start by plotting the y-intercept, as it is a fixed point where the line touches the y-axis. From this point, use the slope to determine the direction and steepness of the line. In our example, the slope \( \frac{2}{3} \) means for every 3 units you move horizontally, the line moves up 2 units.
Once another point is determined using the slope, draw a straight line through the points, extending across the coordinate plane. This line represents every possible solution to the linear equation.
Coordinate Plane
The coordinate plane is a two-dimensional space where we can graph and visualize equations. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis.
Each point on the plane is represented as a pair \( (x, y) \), which indicates its position relative to these axes. For graphing linear equations, the y-intercept is where the line crosses the y-axis, and it's often our starting point on the graph.
Using the slope, we translate the line across the plane, plotting additional points to guide our line. This makes it easier to see solutions visually and understand how changes in the equation affect the line's position and orientation.

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Most popular questions from this chapter

ATHLETICS: Muscle Contraction The fundamental equation of muscle contraction is of the form \((w+a)(v+b)=c\), where \(w\) is the weight placed on the muscle, \(v\) is the velocity of contraction of the muscle, and \(a, b\), and \(c\) are constants that depend upon the muscle and the units of measurement. Solve this equation for \(v\) as a function of \(w, a, b\), and \(c\).

Find, rounding to five decimal places: a. \(\left(1+\frac{1}{100}\right)^{100}\) b. \(\left(1+\frac{1}{10,000}\right)^{10,000}\) c. \(\left(1+\frac{1}{1,000,000}\right)^{1,000,000}\) d. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

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

BUSINESS: Salary A sales clerk's weekly salary is \(\$ 300\) plus \(2 \%\) of her total week's sales. Find a function \(P(x)\) for her pay for a week in which she sold \(x\) dollars of merchandise.

The intersection of an isocost line \(w L+r K=C\) and an isoquant curve \(K=a L^{b}\) (see pages 18 and 32 ) gives the amounts of labor \(L\) and capital \(K\) for fixed production and cost. Find the intersection point \((L, K)\) of each isocost and isoquant. [Hint: After substituting the second expression into the first, multiply through by \(L\) and factor.] $$ 5 L+4 K=120 \text { and } K=180 \cdot L^{-1} $$
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