Problem 13 - Technical Mathematics with Calculus

Question

Rewrite each equation in explicit form and graph for integer values of \(x\) from -3 to 3 . $$x+y-5=0$$

Step-by-Step Solution

Answer

The equation in explicit form is \(y = -x + 5\). The points plotted for \(x\) values from -3 to 3 would be (-3,8), (-2,7), (-1,6), (0,5), (1,4), (2,3), and (3,2), creating a straight line when connected.

Step 1: Solve for y to put in explicit form

To rewrite the equation in explicit form, isolate y on one side of the equation. Starting with the equation: \(x + y - 5 = 0\), add \(-x + 5\) to both sides to get \(y = -x + 5\).

Step 2: Create a table of values

Set up a table with two columns, one for integer values of \(x\) from -3 to 3, and another for corresponding values of \(y\) using the equation \(y = -x + 5\). Calculate \(y\) for each value of \(x\) and fill in the table.

Step 3: Plot the points on a graph

On a coordinate plane, plot the points found in Step 2. Each point will have coordinates of the form \((x, y)\).

Step 4: Draw the line

Connect the points plotted on the graph with a straight line since the relationship between x and y is linear according to the equation \(y = -x + 5\).

Key Concepts

Explicit Form

Understanding the explicit form of a linear equation is crucial for graphing. The explicit form of a linear equation is when the equation is solved for one variable, typically 'y', in terms of the other variable, 'x'. This form is often stated as, for example, 'y = mx + b', where 'm' represents the slope and 'b' signifies the y-intercept – the point where the line crosses the y-axis.

In the given exercise, we transition from the standard form 'x + y - 5 = 0' to the explicit form 'y = -x + 5', which provides us with a clear view of the slope and y-intercept. The slope here is -1, indicating that for every one unit increase in 'x', 'y' will decrease by one unit. The y-intercept is at point (0, 5), showcasing where the line will cross the y-axis.

Table of Values

A table of values is a practical tool to visualize the relationship between two variables in an equation. It consists of two columns, one for 'x' values and one for 'y' values. When creating a table of values for a linear equation in explicit form, we choose a range of 'x' values and use the equation to find the corresponding 'y' values.

For instance, with our equation 'y = -x + 5', if we input 'x' values ranging from -3 to 3, we will obtain a set of ordered pairs that help us plot the graph of the line. Calculating 'y' for each 'x', our table will guide us in graphing the linear relationship accurately, ensuring that each plotted point reflects the true function described by the equation.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface with horizontal and vertical axes, typically labeled 'x' and 'y' respectively. The plane is divided into four quadrants by these axes, and each point on the plane is determined by an ordered pair (x, y).

The origin, where the axes intersect, has coordinates (0, 0) and serves as a central point for reference when plotting. In graphing a linear equation, the coordinate plane allows us to visualize the equation's graph and understand the relationship between 'x' and 'y' geometrically. Identifying points through the table of values, we can plot them onto the plane and observe the shape and direction of the line they create.

Plotting Points

Plotting points is the action of marking specific locations on the coordinate plane that correspond to ordered pairs (x, y). These points help to form the graph of a function or an equation. When plotting points for a linear equation, it's essential to be precise with the locations to accurately reflect the relationship the equation depicts.

To plot points from our table of values for 'y = -x + 5', start from the origin, move horizontally to the 'x' value, and then vertically to the corresponding 'y' value. After plotting all points, the line that connects them will represent the graph of the equation. The linearity of the equation ensures that these points will line up perfectly in a straight line, demonstrating the consistency of the relationship between 'x' and 'y'.