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Chapter 3: Problem 80

Graph equation. Solve for \(y\) first, when necessary. \(y=-3.5 x+4\)

Short Answer

Expert verified
The graph is a straight line with a slope of -3.5, crossing the y-axis at 4.

Step by step solution

01

Identify the Equation Type

The equation given is in the form of a linear equation, commonly written as \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
02

Identify the Slope and Y-intercept

For the equation \( y = -3.5x + 4 \), the slope \( m \) is -3.5, and the y-intercept \( b \) is 4. This means the line crosses the y-axis at \( y = 4 \).
03

Plot the Y-intercept

On the coordinate plane, plot the point (0,4). This is where the line will intersect the y-axis.
04

Use the Slope to Plot a Second Point

Starting from the y-intercept (0,4), use the slope \( -3.5 \). A slope of \(-3.5\) means you go down 3.5 units and 1 unit to the right from the y-intercept. Plot this second point.
05

Draw the Line

Draw a straight line through the points (0,4) and the point you plotted in Step 4. Extend the line in both directions, ensuring it is straight.

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Key Concepts

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

Understanding the Slope
In the world of linear equations, the slope plays a crucial role. The slope describes how steep a line is. It's often represented by the letter \( m \). When interpreting the slope from the equation \( y = mx + b \), it indicates the rate of change of \( y \) with respect to \( x \).

For the equation \( y = -3.5x + 4 \), the slope is \(-3.5\). But what does this number mean?
  • A negative slope, like \(-3.5\), indicates that the line is decreasing. This means as you move from left to right along the graph, the line goes downwards.
  • The value \(3.5\) tells us the exact steepness; for every step right along the x-axis, the line drops 3.5 units.
Interpreting the slope is vital for understanding the behavior of the linear equation, especially when graphing it.
Demystifying the Y-intercept
The y-intercept is where the magic begins when you start plotting your linear equation on a coordinate plane. It is represented by the symbol \( b \) in the linear equation format \( y = mx + b \).

For the given equation \( y = -3.5x + 4 \), the y-intercept is \(4\). But what exactly does the y-intercept mean and why is it so important?
  • The y-intercept gives you the point \((0, b)\) where the line crosses the y-axis. For this equation, the line crosses the y-axis at \((0, 4)\).
  • This point is crucial because it serves as a starting point for drawing the line on a graph.
Understanding the y-intercept helps you to set a basis for determining the line's behavior across the coordinate plane.
Navigating the Coordinate Plane
The coordinate plane is your canvas for graphing linear equations. It's a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

When plotting a linear equation like \( y = -3.5x + 4 \), the coordinate plane helps visualize the relationship between \( x \) and \( y \).
  • Start by identifying the y-intercept and marking it on the y-axis. For \( y = -3.5x + 4 \), plot the point \((0, 4)\).
  • Use the slope to determine the movement from the y-intercept. With a slope of \(-3.5\), drop 3.5 units for each step right.
  • Plot another point using the slope, then draw a line through both points. Extend the line in both directions to cover more of the plane.
The coordinate plane serves as the perfect backdrop for graphing, showing how linear equations create lines that stretch infinitely in both directions.

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Most popular questions from this chapter

Lightning. The function \(D(t)=\frac{t}{5}\) gives the approximate distance in miles that you are from a lightning strike, where \(t\) is the number of seconds between seeing the lightning and hearing the thunder. Find \(D(5)\) and explain what it means.

Halloween Candy. A candy maker wants to make a 60 -pound mixture of two candies to sell for $$ 2\( per pound. If black licorice bits sell for \) 1.90 per pound and orange gumdrops sell for $$ 2.20$ per pound, how many pounds of each should be used?

Production Costs. A television production company charges a basic fee of \(\$ 5,000\) and then \(\$ 2,000\) an hour when filming a commercial. a. Write a linear equation that describes the relationship between the total production costs \(c\) and the hours \(h\) of filming. b. Use your answer to part a to find the production costs if a commercial required 8 hours of filming

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((2,4)\) and \((-1,-1)\) \((8,0)\) and \((11,5)\)

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 4 x-2 y=6 $$
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