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Chapter 3: Problem 74

(a) find three solutions of the equation. (b) graph the equation. \(y=\frac{2}{3} x-4\)

Short Answer

Expert verified
Three solutions are (0, -4), (3, -2), and (-3, -6). Graph the line using these points.

Step by step solution

01

- Find the first solution

To find the first solution, choose a value for x and substitute it into the equation. For example, let's choose x = 0:\[ y = \frac{2}{3} \cdot 0 - 4 \]\[ y = -4 \]So the first solution is (0, -4).
02

- Find the second solution

Choose another value for x and substitute it into the equation. Let's choose x = 3:\[ y = \frac{2}{3} \cdot 3 - 4 \]\[ y = 2 - 4 \]\[ y = -2 \]So the second solution is (3, -2).
03

- Find the third solution

Choose yet another value for x and substitute it into the equation. Let's choose x = -3:\[ y = \frac{2}{3} \cdot (-3) - 4 \]\[ y = -2 - 4 \]\[ y = -6 \]So the third solution is (-3, -6).
04

- Graph the equation

To graph the equation, plot the points (0, -4), (3, -2), and (-3, -6) on a coordinate plane. Draw a straight line through these points, as the equation represents a linear function. The line will extend infinitely in both directions.

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Key Concepts

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

finding solutions
Finding solutions of a linear equation means identifying pairs of x and y values that make the equation true. In this problem, the equation is \(y=\frac{2}{3}x-4\). To find solutions, you can choose any value of x, substitute it into the equation, and solve for y. Here鈥檚 a simple way to do it:
  • Choose a value for x. For instance, if you pick x = 0, you substitute it into the equation, giving you y = -4. So one solution is (0, -4).
  • Choose another value for x, say x = 3. Substituting x = 3 into the equation, you solve for y to get -2. Thus, (3, -2) is another solution.
  • Pick another value for x, such as x = -3. Substitute this value into the equation and solve for y, getting -6. So, (-3, -6) is a third solution.
By substituting different x values and solving for y, you can find as many solutions as you need. This process is helpful since each pair (x, y) you find forms a point that you can plot on a graph.
graphing linear equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through these points. For the equation \(y=\frac{2}{3}x-4\), we already found three solutions: (0, -4), (3, -2), and (-3, -6). The steps to graph this equation are as follows:
  • First, draw a coordinate plane. This consists of a horizontal x-axis and a vertical y-axis intersecting at the origin (0,0).
  • Next, plot each of the solutions on the plane. For instance, plot the point (0, -4) by starting at the origin and moving 0 units along the x-axis and -4 units along the y-axis.
  • Repeat this for the points (3, -2) and (-3, -6).
  • Once you have plotted all points, use a ruler to draw a straight line that passes through them. This line represents the equation \(y=\frac{2}{3}x-4\).
Linear equations always graph to straight lines. The line extends infinitely in both directions, meaning it continues beyond the points you have plotted.
coordinate plane
A coordinate plane is a two-dimensional surface on which you can plot points, lines, and curves. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Here鈥檚 a quick guide to understanding the coordinate plane:
  • The point where the x-axis and y-axis intersect is called the origin, labeled (0,0).
  • Points on the plane are written as pairs (x, y), where x is the horizontal coordinate, and y is the vertical coordinate.
  • The plane is divided into four quadrants. The upper right is Quadrant I (both x and y positive), the upper left is Quadrant II (x negative, y positive), the lower left is Quadrant III (both x and y negative), and the lower right is Quadrant IV (x positive, y negative).
By understanding how to navigate the coordinate plane, you can easily plot points and graph equations. In our example, the points (0, -4), (3, -2), and (-3, -6) each had specific locations that we plotted to create the graph of the linear equation. Mastery of the coordinate plane is fundamental to working with graphs and equations efficiently.

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