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Chapter 5: Problem 32

Sketch the line with the given slope and \(y\) -intercept. $$m=-0.3,(0,-1.4)$$

Short Answer

Expert verified
The line can be sketched using the slope-intercept form: \( y = -0.3x - 1.4 \), with a line through points \( (0, -1.4) \) and \( (1, -1.7) \).

Step by step solution

01

Understanding the Question

We need to sketch the line that has a slope of \( m = -0.3 \) and a \( y \)-intercept of \( (0, -1.4) \). This means the line will cross the \( y \)-axis at \( (0, -1.4) \).
02

Slope-Intercept Form

The equation of a line in slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. For our line, \( y = -0.3x - 1.4 \).
03

Plot the Y-Intercept

On a graph, plot the \( y \)-intercept point \( (0, -1.4) \). This is where the line crosses the \( y \)-axis.
04

Use the Slope

The slope \( m = -0.3 \) means that for each unit increase in \( x \), \( y \) decreases by 0.3 units. From point \( (0, -1.4) \), find a second point by moving one unit to the right along the \( x \)-axis and then 0.3 units down to \( (1, -1.7) \).
05

Draw the Line

Using the points \( (0, -1.4) \) and \( (1, -1.7) \), draw a line through these points to accurately represent the line on the graph.

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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 straightforward way to represent a linear equation. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. This form is very useful because once you identify these two components鈥攁 slope \( m \) and a y-intercept \( b \)鈥攜ou can easily sketch the graph of the line.

  • The slope \( m \) indicates how steep the line is and the direction it tilts. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.

  • The y-intercept \( b \) is simply the point where the line crosses the y-axis. This is always at an x-coordinate of 0.

By using this structure, you can quickly translate a real-world problem into a graphable line鈥攁nd vice versa, allowing you to visually understand relationships between variables.
Y-Intercept
The y-intercept is a key component in the slope-intercept form of a linear equation. It is represented by \( b \) in the equation \( y = mx + b \). To find the y-intercept on a graph, simply look for the point where the line crosses the y-axis. This will always be at an x-coordinate of 0.

For example, in our exercise, the y-intercept is given as \((0, -1.4)\). This means if you were to plot this on a graph, you would place a point directly on the y-axis at -1.4. This point offers a starting location for sketching the rest of your line.
  • The y-intercept tells us the value of \( y \) when \( x = 0 \).

  • Knowing the y-intercept is crucial for setting up your graph correctly, as it anchors the line in space.

Always start plotting your line from this y-intercept point to ensure accuracy in your graph.
Plotting Points on a Graph
Plotting points on a graph is an essential skill for visualizing linear equations. Begin by marking the y-intercept. In our example, that's the point \( (0, -1.4) \). This spot marks where your line will intersect with the y-axis.

Next, you use the slope to determine the next point. A slope of \( -0.3 \) tells you how to move from the y-intercept to find another point:
  • Move 1 unit horizontally across the x-axis (to the right).

  • Since our slope is negative, move 0.3 units vertically down.

  • Connect these two points: \( (0, -1.4) \) and \( (1, -1.7) \).

Placing these points and drawing a straight line through them allows you to visualize the equation as a graph. This line will extend infinitely in both directions, representing all possible solutions to the equation.

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Most popular questions from this chapter

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. A small isolated farm uses a windmill and a gas generator for power. During a 10 -day period, they produced \(3010 \mathrm{kW}\) \cdoth of power, with the windmill operating at \(45.0 \%\) of capacity and the generator at capacity. During the following 10-day period, they produced 2900 kW - h with the windmill at \(72.0 \%\) of capacity and the generator down \(60 \mathrm{h}\) for repairs (at capacity otherwise). What is the capacity (in \(\mathrm{kW}\) ) of each?

Answer the given questions about the determinant to the right. $$\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|$$ What change in value occurs if both the rows and the columns are interchanged?

Solve the given systems of equations by determinants. $$\begin{aligned} &2 x-3 y=4\\\ &2 x+y=-4 \end{aligned}$$

Solve the given systems of equations algebraically. The numbers are approximate. $$\begin{aligned} &3.6 x+5.2 y-z=-2.2\\\ &3.2 x-4.8 y+3.9 z=8.1\\\ &6.4 x+4.1 y+2.3 z=5.1 \end{aligned}$$

Solve the given systems of equations by determinants. $$\begin{aligned} &12 t+9 y=14\\\ &6 t=7 y-16 \end{aligned}$$
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