# Linear Graphs

When we have an equation with two different unknowns, like y = 2x + 1, we cannot solve the equation.

We can instead find pairs of x and y values that make the left side equal the right side

We can use a table of values to show the number pairs:

 x 0 1 2 3 y

To complete this table of values, for y = 2x + 1, we need to find out the value of y when x = 0, when x = 1, when x = 2 and when x = 3

We do this using substitution

When x = 0
y = 2(0) + 1
y = 1

When x = 1
y = 2(1) + 1
y = 3

When x = 2
y = 2(2) + 1
y = 5

When x = 3
y = 2(3) + 1
y = 7

We can use these values to complete the table

 x 0 1 2 3 y 1 3 5 7

Example 1: Complete the table of values for x + y = 6

 x 0 1 2 3 y

We can find the y values using substitution:

When x = 0
0 + y = 6
y = 6

When x = 1
0 + y = 6
y = 5

When x = 2
0 + y = 6
y = 4

When x = 3
0 + y = 6
y = 3

We can now complete the table:

 x 0 1 2 3 y 6 5 4 3

Example 2: Complete the table of values for y = 3x - 5

 x -3 -2 -1 0 1 2 3 y

In this example we have negative x values in the table. It is often easier to start with the positive x values:

When x = 3
y = 3(3) - 5
y = 4

When x = 2
y = 3(2) - 5
y = 1

When x = 1
y = 3(1) - 5
y = -2

When x = 0
y = 3(0) - 5
y = -5

 x -3 -2 -1 0 1 2 3 y -5 -2 1 4

We can notice a pattern in these y values, they are going up by 3 each time (from left to right or down by 3 each time from right to left).

We can expect that for x = -1, y = -8
When x = -2, y = -11
and when x = -3, y = -14

Substituting gives:

When x = -1
y = 3(-1) - 5
y = -8

When x = -2
y = 3(-2) - 5
y = -11

When x = -3
y = 3(-3) - 5
y = -14

If we make a mistake in our calculations we will be able to notice as our values would not fit the pattern.

 x -3 -2 -1 0 1 2 3 y -14 -11 -8 -5 -2 1 4

Try these:

We can use a table of values to draw a graph. Each pair of values become a set of coordinates (x,y).

Example: Here is the table of values for y = 2x - 3
draw the graph for y = 2x - 3

 x -3 -2 -1 0 1 2 3 y -9 -7 -5 -3 -1 1 3

On a graph we will plot the coordinates:
(-3,-9), (-2,-7), (-1,-5), (0,-3), (1,-1), (2,1) and (3,3)

We join the points up with a straight line.

Example: Draw the graph for y = 3x + 1

Here we have not been given a table of values.

We can create a table of values. We can see that the x values on the graph are -2, -1, 0, 1 and 2. We can use these for our table of values:

 x -2 -1 0 1 2 y

We can now complete our table bu substituting our x values into the equation:

When x = 2
y = 3(2) + 1
y = 7

When x = 1
y = 3(1) + 1
y = 4

When x = 0
y = 3(0) + 1
y = 1

When x = -1
y = 3(-1) + 1
y = -2

When x = -2
y = 3(-2) + 1
y = -5

We can use these values to complete the table

 x -2 -1 0 1 2 y -5 -2 1 4 7

The next step is to plot the coordinates:
(-2,-5), (-1,-2), (0,1), (1,4) and (2,7)

And finally we use a ruler to draw the line