Problem 77 - 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. $$y=x+\frac{1}{2}$$

Step-by-Step Solution

Answer

The solutions for the equation \(y = x + \frac{1}{2}\) are (-2, -3/2), (-1, -1/2), (0, 1/2), (1, 3/2), (2, 5/2). The corresponding graph is a line slanting upwards, intercepting y-axis at \(\frac{1}{2}\)

Step 1 - Understanding the equation

The linear equation is in the form of \(y = mx + c\) where \(m\) is the slope of the line and \(c\) is the y-intercept. Here \(m=1\) and \(c= \frac{1}{2}\). This indicates a line with a slope of 1 that crosses the y axis at \(\frac{1}{2}\).

Step 2 - Generating values

A table of values can be generated by substituting various numbers for \(x\) and then solving the equation for \(y\). Let's solve for \(y\) when \(x\) is -2, -1, 0, 1, 2.

Step 3 - Calculating values

Find the coresponding y values for the x values listed above.\n When \(x = -2\), \(y = -2 + \frac{1}{2} = -\frac{3}{2}\)\n When \(x = -1\), \(y = -1 + \frac{1}{2} = -\frac{1}{2}\)\n When \(x = 0\), \(y = 0 + \frac{1}{2} = \frac{1}{2}\)\n When \(x = 1\), \(y = 1 + \frac{1}{2} = \frac{3}{2}\)\n When \(x = 2\), \(y = 2 + \frac{1}{2} = \frac{5}{2}\),

Step 4 - Graphing the linear equation

Use a graph paper, mark the five values calculated in the previous step (-2, -3/2), (-1, -1/2), (0, 1/2), (1, 3/2), (2, 5/2) on the graph paper in the Cartesian coordinate system. Connect these points to draw the line.

Key Concepts

Linear Equation in Two Variables

A linear equation in two variables looks like a simple algebraic statement that describes a straight line when graphed on a Cartesian plane. It typically takes the form of \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) indicates the y-intercept, the point where the line crosses the y-axis.

In the given exercise, the equation \(y = x + \frac{1}{2}\) is an example of a linear equation where the slope (\(m\)) is 1 and the y-intercept (\(c\)) is \(\frac{1}{2}\). Understanding these components is crucial because they dictate the angle at which the line tilts and where it slices through the y-axis.

Table of Values

Constructing a table of values is a systematic method to solve a linear equation. By choosing various numbers for one variable (usually \(x\)), we can determine the corresponding values of the other variable (\(y\)). This process of substitution reveals ordered pairs (\(x, y\)) that satisfy the given equation.

For our equation, a series of \(x\) values is selected, often including negative values, zero, and positive values, to get a full picture of the line's behavior. As illustrated in the solution, values for \(x\) have been chosen from -2 to 2. The resulting y values form pairs with their respective x values, which are then plotted as points on the graph to eventually draw the line.

Slope and Y-intercept

The slope of a line is a measure of its steepness and is calculated as the rise over the run between any two points on the line. In the equation \(y = mx + c\), the slope is represented by \(m\). A positive slope means the line ascends from left to right, while a negative slope indicates it descends. A slope of zero implies a horizontal line, and undefined slope indicates a vertical line.

The y-intercept \(c\) is straightforward; it's simply the y-coordinate of the point where the line intersects the y-axis. In our example, the slope is 1, which signifies an angle of 45 degrees, and the y-intercept is \(\frac{1}{2}\), positioning the starting point of the line just above the origin.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane composed of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at a point called the origin. Each point on this plane can be specified by an ordered pair of numbers \((x, y)\), known as coordinates.

When graphing a linear equation, we plot the series of points from our table of values onto this plane. Each point is found by starting at the origin, moving along the x-axis by the first number, and then vertically by the second number in the pair. Connecting these plotted points as shown in the solution gives us the visual representation of the equation as a line on the grid.