Problem 32 - Introductory Algebra for College Students

Question

Graph each linear equation using the slope and y-intercept. $$y=\frac{1}{3} x+2$$

Step-by-Step Solution

Answer

The graph of the equation \(y = \frac{1}{3} x + 2\) would be a straight line that passes through the points (0,2) and (3,3).

Step 1: Identify the slope and y-intercept

For the linear equation \(y = \frac{1}{3} x + 2\), the slope, 'm' is \frac{1}{3} and the y-intercept 'b' is 2.

Step 2: Plot the y-intercept

Start by plotting the y-intercept, b(=2) on the y-axis. This is the point where the line crosses the y-axis.

Step 3: Use the slope to find another point and draw the line

Starting from the y-intercept, move in the direction specified by the slope to find a second point. Since the slope is positive \( \frac{1}{3} \), move up 1 unit (the numerator of the slope fraction) and right 3 units (the denominator of the slope fraction) to obtain another point on the line. Then draw a straight line through these two points, which is the graph of the given equation.

Key Concepts

Slope and Y-Intercept

Understanding the slope and y-intercept is crucial when dealing with linear equations. The slope, often represented as the letter 'm', determines the steepness and direction of the line. If 'm' is positive, the line tilts upwards; if negative, it tilts downwards. To find the slope from an equation like \(y = \frac{1}{3} x + 2\), you look at the coefficient of 'x', which is \(\frac{1}{3}\) in this case.

Meanwhile, the y-intercept is the point where the line crosses the y-axis and is typically denoted as 'b'. In the equation, it's the constant term, which is 2 here. This point can be easily plotted and is (0,2) on the coordinate plane. Together, the slope and y-intercept form the starting point for graphing the entire line.

Key Takeaways:

• The slope indicates the direction and steepness of the line.
• The y-intercept is where the line crosses the y-axis.
• In the equation \(y = mx + b\), 'm' is the slope and 'b' is the y-intercept.

Linear Equation Graph

A linear equation graph is a visual representation of all the possible solutions to a linear equation. It's always a straight line. Each point on this line is a solution of the equation. In the graph of \(y = \frac{1}{3} x + 2\), for example, every point on the line satisfies the relation between x and y that the equation provides.

Graphs aid in understanding the relationship between dependent and independent variables, which are represented on the y and x-axes, respectively. They are also useful in predicting values and solving systems of equations. By plotting lines from different equations on the same set of axes, you can visually find intersections, which represent solutions common to all the equations involved.

Visualizing Equations:

• A line represents all solutions to the equation.
• It shows relationships between variables.
• Useful for predictions and solving systems of equations.

Plotting Linear Equations

Plotting a linear equation involves translating the information given by its mathematical expression onto a two-dimensional plane. To plot an equation like \(y = \frac{1}{3} x + 2\), you begin by marking the y-intercept on the y-axis. Here, you'd plot a point at (0, 2).

Next, you use the slope to find another point. Starting from the y-intercept, the slope \(\frac{1}{3}\) tells you to rise one unit and run three units to the right. This movement from the y-intercept gives you the second point, after which you can draw a line through the points to extend it infinitely in both directions.

Remember that any two points can determine a line, but using the y-intercept and the slope ensures accuracy and efficiency when plotting.

Steps for Plotting:

• Plot the y-intercept on the coordinate plane.
• From there, use the slope to find a second point.
• Draw a straight line through the points, extending it across the plane.