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Chapter 3: Problem 55

Graph each equation. $$y-3.5=0$$

Short Answer

Expert verified
The graph of the equation \(y - 3.5 = 0\) is a straight horizontal line passing through the y-axis at y = 3.5.

Step by step solution

01

Rewrite the equation

First thing to do is simplify the equation to a more familiar format. In this case, \(y - 3.5 = 0\) can be simplified to \(y = 3.5\)
02

Determine the co-ordinates

The equation \(y = 3.5\) means that regardless of the \(x\) value, \(y\) will always be 3.5. So some possible coordinates can be \( (0, 3.5), (1, 3.5), (-1, 3.5), (2, 3.5) \) and so on.
03

Graph the equation

Now, plot these points on a graph. Since \(y\) is constant for all \(x\), the points will form a straight horizontal line which passes through all the points on the y-axis at y=3.5

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Key Concepts

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

Horizontal Lines in Linear Equations
When encountering linear equations that represent horizontal lines, it's essential to recognize a key characteristic: the y-value is constant while the x-value can vary freely. In the equation given, \(y = 3.5\), the horizontal line signifies that no matter what number you choose for x, the y-value will always be 3.5.

Visualizing this on a graph, imagine standing at the point (0, 3.5) on a road that runs perfectly east to west; you can move as far in either direction as you like but your elevation remains the same. This is what plotting a horizontal line in a linear equation looks like鈥攁n infinite path with unchanging elevation.
Coordinate Plane
The coordinate plane is the stage on which we plot our equations, consisting of a horizontal number line (the x-axis) and a vertical number line (the y-axis), intersecting at the origin (0,0).

It's like a map that helps us find the location of points defined by pairs of numbers, known as coordinates, which tell us how far along the x-axis and how high up the y-axis a point is. In our exercise, the points we plot all share the same y-value as they lie on a horizontal line parallel to the x-axis.
Algebraic Equations
Becoming familiar with algebraic equations is like learning a new language in mathematics. These equations are structured relationships using numbers, variables, and operators that show how two things are equal. In the equation \(y = 3.5\), it's a simple declaration: y is always equal to 3.5, regardless of x's value.

Common in solving algebraic problems is performing operations or manipulations to isolate variables or to express the equation in a simpler form. Here, it's already simplified, demonstrating the essence of horizontal lines without further actions needed.
Plotting Points
The process of plotting points is like placing a pin on a board to mark a location. When working with linear equations, we select pairs of numbers, each called an (x, y) coordinate, which correspond to the values specified by the equations.

In our example, the coordinates might be (0, 3.5), (1, 3.5), and so on. This means you move 0 units along the x-axis and 3.5 units up the y-axis for the first point, 1 unit along the x-axis and 3.5 units up on the y-axis for the second, and you continue similarly for additional points. Plotting these correctly is vital as it helps us to visualize the relationships represented by the algebraic equations.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for the given ordered pairs \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left(x_{1}, y_{1}\right)=(3,4) ;\left(x_{2}, y_{2}\right)=(5,4)$$

a. Graph each of the following points: $$\left(1, \frac{1}{2}\right),(2,1),\left(3, \frac{3}{2}\right),(4,2)$$ Parts (b)-(d) can be answered by changing the sign of one or both coordinates of the points in part (a). b. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the \(y\) -axis of your graph in part (a)? c. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the \(x\) -axis of your graph in part (a)? d. What must be done to the coordinates so that the resulting graph is a straight-line extension of your graph in part (a)?

Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((2,4)\) and has the same \(y\) -intercept as the line whose equation is \(x-4 y=8\)

A new car worth 45,000 dollars is depreciating in value by 5000 dollars per year. The mathematical model $$y=-5000 x+45,000$$ describes the car's value, \(y,\) in dollars, after \(x\) years. a. Find the \(x\) -intercept. Describe what this means in terms of the car's value. b. Find the \(y\) -intercept. Describe what this means in terms of the car's value. c. Use the intercepts to graph the linear equation. Because \(x\) and \(y\) must be nonnegative (why?), limit your graph to quadrant I and its boundaries. d. Use your graph to estimate the car's value after five years.

If two lines are parallel, describe the relationship between their slopes.
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