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Chapter 3: Problem 67

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=\frac{1}{2} x$$

Short Answer

Expert verified
The solutions to the equation \(y=\frac{1}{2}x\) are \((-4, -2)\), \((-2, -1)\), \((0, 0)\), \((2, 1)\), and \((4, 2)\), and its graph is a straight line that passes through these points.

Step by step solution

01

Compute Solutions

Substitute values into the equation \(y=\frac{1}{2}x\) to find at least five solutions. For instance, when \(x = -4\), \(x = -2\), \(x = 0\), \(x = 2\) and \(x = 4\), compute for the corresponding 'y' values.
02

Create the Table of Values

The table is comprised of 'x' values and their corresponding 'y' values. The five solution pairs are \((-4, -2)\), \((-2, -1)\), \((0, 0)\), \((2, 1)\), and \((4, 2)\).
03

Plot the Solutions on the Graph

Draw the x and y axes on your graph. Then plot the ordered pairs \((-4, -2)\), \((-2, -1)\), \((0, 0)\), \((2, 1)\), and \((4, 2)\) on the graph. Each pair represents points on the line.
04

Draw the Line of the Equation

After plotting the solutions, connect them with a straight line. This line represents the solutions to the equation \(y=\frac{1}{2}x\). Every point on this line is a solution to the equation.

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Key Concepts

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

Graphing Equations
Graphing a linear equation involves visualizing how the solutions to the equation appear on a coordinate plane. Linear equations in two variables, like the one given, form straight lines when graphed. The equation \( y = \frac{1}{2}x \) means for every value of \( x \), \( y \) is one-half of \( x \). This consistent relationship between \( x \) and \( y \) is what produces the straight line on the graph.
  • Start by understanding the format of the equation, in this case a slope-intercept form where \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept.
  • The slope, \( \frac{1}{2} \), tells us the line rises 1 unit for every 2 units it moves to the right.
  • The line crosses the origin because there's no \( b \) term in the equation, meaning \( b = 0 \).
This simple equation gains physical meaning on the graph as you plot the points related to it, showing a clear path or trend in the data.
Table of Values
Creating a table of values helps to organize and keep track of your computations. It also visually lays out the relationship between \( x \) and \( y \). For our equation, \( y = \frac{1}{2}x \), carefully choosing values for \( x \) will lead to corresponding \( y \) values that can be plotted to graph the line.
  • Select at least five distinct values for \( x \). It's useful to include both negative and positive values, as well as zero.
  • Calculate the corresponding \( y \) value for each selected \( x \) by substituting them into the equation.
  • Record each \( x \) and \( y \) pair in your table. The table might look like this:
    \(\begin{array}{c|c}x & y \\hline-4 & -2 \-2 & -1 \0 & 0 \2 & 1 \4 & 2 \\end{array}\)
Using the table of values clarifies the linear nature of the equation and provides the necessary data for plotting the points on a graph.
Solution Pairs
Solution pairs are the \( (x, y) \) coordinates that satisfy the equation. Think of them as ordered pairs that tell you exactly where to place a point on the graph. For a linear equation such as \( y = \frac{1}{2}x \), each solution pair lies on the line represented by the equation.
  • Identify several values for \( x \) and calculate the corresponding \( y \) values to get solution pairs like \((-4, -2)\), \((-2, -1)\), \((0, 0)\), \((2, 1)\), and \((4, 2)\).
  • Each pair indicates specific points that satisfy the linear equation.
  • These pairs help visualize the relationship described by the equation; plotting them should form a straight line.
Understanding solution pairs is vital for graphing because they are direct results of the equation, and are the exact coordinates where the line crosses through on the graph.
Plotting Points
Plotting points on a graph takes the outputs of your equation and translates them into a visual format on the coordinate plane. It is a crucial step in graphing as it provides a foundation for drawing the line of the equation.
  • Begin by drawing both the x-axis and y-axis on the graph paper. Label each axis with suitable increments to ensure all your points fit your graph.
  • For each solution pair from your table (e.g., \((-4, -2)\), \((-2, -1)\), \((0, 0)\)), plot them on the graph by locating the x-value on the x-axis and finding the corresponding y-value on the y-axis.
  • Mark a dot where each \( (x, y) \) pair intersects. Doing this accurately reflects each point as a place on your line.
By plotting these points and then connecting them, you draw the line which visually represents the full set of solutions for your linear equation.

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Most popular questions from this chapter

Find the quotient: \(\frac{2}{3} \div\left(-\frac{5}{4}\right)\)

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