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Chapter 7: Problem 35

Choose a slope and \(y\) -intercept. Write an equation and then graph the line.

Short Answer

Expert verified
The equation is \( y = 2x - 3 \), with a graph starting at \( (0, -3) \) and passing through \( (1, -1) \).

Step by step solution

01

Choose a Slope

Let's choose a slope for the line, which is usually represented by the letter \( m \). For simplicity, let's select \( m = 2 \). This means that for every 1 unit increase in \( x \), \( y \) increases by 2 units.
02

Choose a Y-Intercept

The \( y \)-intercept is where the line crosses the \( y \)-axis. This is represented by \( b \). Let's choose \( b = -3 \). This means that when \( x = 0 \), \( y = -3 \).
03

Write the Equation of the Line

Now that we have a slope \( m = 2 \) and a \( y \)-intercept \( b = -3 \), we can write the equation of the line in the slope-intercept form \( y = mx + b \). Thus, the equation is \( y = 2x - 3 \).
04

Graph the Line

To graph the equation \( y = 2x - 3 \), start by plotting the \( y \)-intercept on the graph at point \( (0, -3) \). Then, use the slope \( m = 2 \) to find another point. From \( (0, -3) \), move up 2 units and to the right 1 unit to reach the next point \( (1, -1) \). Plot this point and draw a line through these two points, extending it in both directions.
05

Verify the Line

Check that the points \( (0, -3) \) and \( (1, -1) \) satisfy the equation \( y = 2x - 3 \). For the point \( (0, -3) \), substitute \( x = 0 \) into the equation: \( y = 2(0) - 3 = -3 \), which is correct. For \( (1, -1) \), substitute \( x = 1 \):\( y = 2(1) - 3 = -1 \), which is also correct. Thus, the line has been correctly graphed.

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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 one of the most popular ways to express the equation of a straight line. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) stands for the \( y \)-intercept. This format is incredibly straightforward, making it easy to understand and use when graphing lines or solving problems involving linear equations.
  • **Slope (\( m \)):** This is a measure of the steepness of the line. A larger absolute value indicates a steeper line.
  • **Y-Intercept (\( b \)):** This is the point where the line crosses the \( y \)-axis. It provides an easy starting point for graphing the line.
The slope-intercept form is handy for quickly sketching graphs or understanding how changes in \( m \) and \( b \) impact the line's position and angle.
Graphing Lines
Graphing linear equations is a simple process that starts by identifying the \( y \)-intercept and slope from the slope-intercept form equation. Here's a breakdown of how to graph using this form:
  • **Start with the \( y \)-intercept:** Locate the \( y \)-intercept on the graph, which is where the graph cuts through the \( y \)-axis. This provides a base point from which to start.
  • **Use the Slope:** From the \( y \)-intercept, use the slope to find another point. A slope of \( m = 2 \) indicates that for every 1 unit moved along the \( x \)-axis, the \( y \) value increases by 2 units. This makes it easy to plot additional points quickly.
Once you've marked at least two points, draw a line through them, extending it in both directions. This line represents the solution set of the equation. The elegant part of linear graphs is their simplicity, allowing quick visualization of relationships between variables.
Y-Intercept
The \( y \)-intercept is a crucial aspect of linear equations and graphing. It gives you the starting point of the line on the graph before considering the slope. When we say a line crosses the \( y \)-axis at \( b \), it means that when \( x = 0 \), the value of \( y \) is \( b \). In practical terms:
  • If \( b \) is positive, the line will cross the \( y \)-axis above the origin.
  • If \( b \) is zero, the line passes through the origin itself.
  • If \( b \) is negative, the line intersects the \( y \)-axis below the origin.
Recognizing the \( y \)-intercept is especially useful when you start graphing because it provides an initial position for placing your line. It sets the stage for applying the slope to get the full picture of the line's path across the graph.

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Most popular questions from this chapter

State the slope and the \(y\) -intercept of the graph of each equation. $$5 x+4 y=20$$

Which statement is true about the data in the table? A The data represent a function. B The data do not represent a function. C As the value of \(x\) increases, the value of \(y\) increases. D A graph of the data would not pass the vertical line test. $$\begin{array}{|c|c|}\hline x & y \\\\\hline-4 & -4 \\\\\hline 2 & 16 \\\\\hline 5 & 8 \\\\\hline 10 & -4 \\\\\hline 12 & 15 \\ \hline\end{array}$$

Write an equation in slope-intercept form for each table of values. $$\begin{array}{|r|r|r|r|r|} \hline x & -1 & 0 & 1 & 2 \\ \hline y & -7 & -3 & 1 & 5 \\ \hline \end{array}$$

Write each number in scientific notation. $$0.00072$$

Use the following information. The altitude in feet \(y\) of a hang glider who is slowly landing can be given by \(y=300-50 x,\) where \(x\) represents the time in minutes. State the slope and \(y\) -intercept of the graph of the equation and describe what they represent.
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