Problem 70 - Introductory Algebra for College Students

Question

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=\frac{1}{4} x$$

Step-by-Step Solution

Answer

The solutions to the equation \(y=\frac{1}{4}x\) are (0,0), (4,1), (8,2), (12,3), and (16,4). The linear equation graph starts from the origin and inclines to the right with a slope of \(1/4\).

Step 1: Construct a Table of Values

We need at least five solutions for this equation. We'll start with \(x=0\) and then progressively increase x by four. We substitute each \(x\) into the equation to find the corresponding \(y\) values. So, the table of values would look like: \n |x | y| |---|----| |0 | 0 | |4 | 1 | |8 | 2 | |12 | 3 | |16 | 4 |

Step 2: Plot the Points

Now that we have our table of values, we can plot these points on a graph. The points we get are (0,0), (4,1), (8,2), (12,3), and (16,4).

Step 3: Connect the Points

Finally, we need to connect these points using a ruler or straight edge. This line is the graph of the equation \(y = \frac{1}{4}x\). It starts from the origin and ascents as we move to the right.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that helps visualize solutions to equations in two variables. A linear equation in two variables is typically expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the equation \(y = \frac{1}{4}x\), the graph is a straight line through the origin, since the y-intercept is \(0\). The slope \( \frac{1}{4} \) tells us that for every 4 units we move along the x-axis, the y-axis increases by 1 unit. To graph, you plot points that are solutions of the equation and connect them. These points lie on a straight line that represents all possible solutions. The graph of a linear equation helps to understand the relationship between two variables.

Tables of Values

Tables of values are a crucial step when graphing equations. They help us easily find solutions that can be plotted on a graph. For any given linear equation like \(y = \frac{1}{4}x\), the table of values consists of paired numbers that satisfy the equation when substituted into it.To create a table of values, choose several values for \(x\) and calculate the corresponding \(y\) values using your equation:

• Start with an initial value, such as \(x = 0\), and find \(y\).
• Continue by systematically choosing other \(x\) values and computing the \(y\) values.
• Your table might look like:
  • \(x = 0\), \(y = 0\)
  • \(x = 4\), \(y = 1\)
  • \(x = 8\), \(y = 2\)
  • \(x = 12\), \(y = 3\)
  • \(x = 16\), \(y = 4\)

Tables of values not only help in graphing but also confirm the linearity of the equation. They show a consistent increment that reflects the equation's slope.

Solutions of Equations

Solutions to equations are the values of variables that satisfy the equation. For a two-variable linear equation, such as \(y = \frac{1}{4}x\), a solution is any pair \((x, y)\) that makes the equation true.When you graph a linear equation, each point on the line represents a solution. For instance, the point \((8, 2)\) is a solution because when you substitute it back into the equation, it holds true: \(2 = \frac{1}{4} \times 8\). Finding solutions using a table of values ensures that you have multiple points which can then be plotted to form the graph. This ensures accuracy in representation and helps in verifying the correctness of the task at hand.

Two-Variable Equations

Two-variable equations involve expressions that have two different variables, commonly \(x\) and \(y\). These equations describe a relationship between the two variables and form straight lines when graphed.For example, \(y = \frac{1}{4}x\) describes a line where the value of \(y\) depends on the value chosen for \(x\). Such a relationship is linear, which means the graph will be a straight line, illustrating how the value of one variable changes in relation to the other. Understanding two-variable equations is essential, as they often appear in real-world problems to model linear relationships, such as rate problems, cost predictions, or time-distance scenarios.By plotting a few calculated points from a table of values, these equations can be visualized, providing insights into the variables' behavior and interaction.