Problem 44 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. $$ x+y=1 $$

Step-by-Step Solution

Answer

The graph of the equation \( x + y = 1 \) results in a straight line passing through the points (-1,2), (0,1), (1,0), and (2,-1).

Step 1: Creating the Table of Values

Start by selecting values for \( x \) and substitute these into the equation to find corresponding values for \( y \). Let's start by choosing \( x = -1, 0, 1, 2 \) respectively.

Step 2: Finding Corresponding 'Y' Values

Substitute the \( x \) values into the equation \( x + y = 1 \) to find the corresponding \( y \) values. When \( x = -1 \), the equation becomes \( -1 + y = 1 \), solving for \( y \) we get \( y = 2 \). Repeat this for all other values of \( x \) to get the corresponding \( y \) values.

Step 3: Creating the Points to Be Plotted on the Graph

After determining the \( y \) values that correspond to the chosen \( x \) values, the points will be: When \( x = -1, y = 2 \) gives (-1,2), \( x = 0, y = 1 \) gives (0,1), \( x = 1, y = 0 \) gives (1,0), and \( x = 2, y = -1 \) gives (2,-1).

Step 4: Plotting the points on the Graph

Plot these points on a graph and draw a line that passes through all the points to represent the equation \( x + y = 1 \).

Key Concepts

Table of Values

When grappling with the task of graphing a linear equation, a table of values proves to be an invaluable tool. This tabular method is a systematic way to organize two sets of numbers that are related to each other by the equation. For the equation \( x+y=1 \), constructing a table of values involves selecting a range of numbers for \( x \) and then solving the equation to find the matching numbers for \( y \).

Students often find it helpful to start with simple numbers, such as \( x = -1, 0, 1, 2 \) because they make the arithmetic straightforward. Once the \( x \) values are chosen, each is substituted into the equation to determine the corresponding \( y \) values. This process creates pairs of numbers which, when plotted, will outline the shape of the graph.

Solving for Y

Solving for \( y \) in a linear equation is a critical skill for graphing. For the equation at hand, \( x + y = 1 \), the next step after setting up the table of values is to isolate \( y \). This maneuver is done by subtracting \( x \) from both sides of the equation.

For instance, if we select \( x = -1 \), the equation transforms into \( -1 + y = 1 \). To solve for \( y \) in this instance, you add \( 1 \) to both sides, resulting in \( y = 2 \). This process should be repeated for each chosen value of \( x \) so that you will have a set of ordered pairs ready to be plotted on the graph.

Plotting Points on a Graph

Plotting the points on a graph is like mapping out treasures; each point represents a piece of the puzzle that is your linear equation. After using the table of values to find out the corresponding \( x \) and \( y \) numbers, you take each pair – \( (x, y) \) – and locate their position on a two-dimensional grid.

For every ordered pair, such as \( (-1, 2) \) or \( (0, 1) \) from our table, plot them accordingly along the \( x \) (horizontal) and \( y \) (vertical) axes of the graph. The point \( (-1, 2) \) means you move one step to the left and two steps upward. After all points are placed on the graph, they should lie in a straight line, revealing the visual representation of the equation.

Linear Equation Representation

A linear equation's graph is a visual representation showing how two variables are related linearly. When plotted, a linear equation such as \( x + y = 1 \) will always form a straight line. This line illustrates all the infinite pairs of \( x \) and \( y \) that satisfy the equation.

Graphing the equation by plotting the points provides a quick and effective way to see solutions at a glance. Moreover, the slope of the line reveals how steep the line is, while the y-intercept (where the line crosses the \( y \) axis) offers a starting point on the graph. To truly understand the concept, practice plotting different linear equations and observe how changes in the equation affect the graph.