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## Introduction

A line can be defined as a straight set of points that extends forever in two directions.  In Algebra, most two-variable equations have graphs that form a line.  One key component of a line is that it is actually comprised of individual points.

Each of the individual points can be found by using the given equation to find a small number of solutions of the equation.  To find a solution, simply find a pair of variables that work correctly for the equation.  For example, consider the equation y = 3x + 1.  Given below is one coordinate that is not a solution of the equation, and a two coordinates that are solutions.

## Lesson

Suppose a parking garage has a fee schedule that is \$3 to order to park plus \$2 per hour.  This fee schedule can be represented by an equation.  However, most people find it easier to understand it when the information is shown as a graph.

Most parking garages have a sign that lists the prices for each hour or portion of an hour.  This particular garage’s sign is shown to the right.

This garage’s fee schedule can be represented by the equation y = 2x + 3, where x stands for the number of hours and y stands for the parking fee.  The comparison between the number of hours and the cost of parking can be shown on a graph.  Before drawing the graph, it makes sense to find the coordinates of 3 or 4 points using a table or t-chart.  Then, place these points on the graph.

Graph of y = 2x + 3:

The data above is a linear function whose graph is a line.  A line is formed because the cost of parking goes up at a constant rate each hour.  Each individual point on the table corresponds to one of the parking prices on the garage sign above.  The graph provides an idea of just steeply the price goes up for each hour.

You can graph any equation by first putting points on a t-chart.  You want to pick points that are convenient, so it is common to pick the first few x-values (x = 0, 1, 2, 3, etc).

Here is the t-chart and graph of the equation y = x + 6:

When dealing with an equation containing fractions, it may make sense to pick x-values other than 0, 1, 2, and 3 in order to make it easier to find individual coordinates.

Consider the equation y = x – 5:

The table and graph above represent this equation.  In the table, it is easier to pick x-values that yield positive or negative whole number y-values when put into the equation.  In this equation, x = 1, x = 2, x = 3, etc. all result in fractional answers.  However x = 0, x = 5, x = 10, and all the multiples of 5 result in whole number values for the resulting variable y.

Of course, you are free to pick any x-values that you like even if they result in fractions.  However, graphing fractions is undesirable for two reasons:

•  Dealing with adding and subtracting fractions is harder and potentially frustrating compared to adding and subtracting whole numbers
• Graphing fractions or mixed numbers requires that you estimate the distance since you must place the point somewhere between two the whole numbers

Each of the equations given so far in this lesson has been in the form y = ___x + ___.  This is called slope-intercept form and is written in a way that is easy to translate into coordinates (and also a graph.)  However, there are many other ways to use equations to compare x and y.

Consider the equation 6x + 3y = 18, which is said to be written in standard form.  Using trial and error, you can find several whole number points that work for this graph.  The points are shown below.

The key to graphing a line is in being able to construct a t-chart.  After doing a few t-charts, you will become more comfortable with the process and start to notice patterns in the t-charts.  Always include several points in your t-chart to give yourself the best chance to draw an accurate graph.

## Try It

Find four coordinates that work for each equation.  Then draw the graph of the equation.

1)  y = x – 1

2)  y = 4x – 3

3)  y = -x + 2

4)  2x + 4y = 16

1)

2)

3)

4)