# Linear Equation Table

## Part I. How Linear Equations relate to Tables Of Values

### Equations as Relationships

The equation of a line expresses a relationship between x and y values on the coordinate plane. For instance, the equation $$y = x$$ expresses a relationship where every x value has the exact same y value. The equation $$y = 2x$$ expresses a relationship in which every y value is double the x value, and $$y = x + 1$$ expresses a relationship in which every y value is 1 greater than the x value.

#### So what about a Table Of Values?

Since, as we just wrote, every linear equation is a relationship of x and y values, we can create a table of values for any line. These are just the $$x$$ and $$y$$ values that are true for the given line. In other words, a table of values is simply some of the points that are on the line.

##### Example 1

Equation: $$\red y = \blue x + 1$$

Table of Values

 $$\blue x \text { value}$$ Equation $$\red y \text{ value}$$ y = x + 1 $$\blue 3$$ $$y = ( \blue 3 ) + 1$$ $$\red 4$$ $$\blue 4$$ y = ($$\blue 4$$ ) + 1 $$\red 5$$ $$\blue 5$$ $$y = (\blue 5 ) + 1$$ $$\red 6$$ $$\blue 6$$ $$y = ( \blue 6) + 1$$ $$\red 7$$
##### Example 2

Equation: y = 3x + 2

Table of Values

 X Value Equation Y value y = 3x + 2 1 y = 3(1) + 2 5 2 y = 3(2) + 2 7 3 y = 3(3) + 2 11 4 y = 3(4) + 2 14

So, to create a table of values for a line, just pick a set of x values, substitute them into the equation and evaluate to get the y values.

### Practice Creating a Table of Values

##### Problem 1

Create a table of values of the equation y = 5x + 2.

Create the table and choose a set of x values.

 X Value Equation Y value y = 5x + 2 1 2 3 4

Substitute each x value (left side column) into the equation.

 X Value Equation Y value y = 5x + 2 1 y = 5(1) + 2 2 y = 5(2) + 2 3 y = 5(3) + 2 4 y = 5(4) + 2

Evaluate the equation (middle column) to arrive at the y value.

 X Value Equation Y value y = 5x + 2 1 y = 5(1) + 2 7 2 y = 5(2) + 2 12 3 y = 5(3) + 2 17 4 y = 5(4) + 2 22

An Optional step, if you want, you can omit the middle column from your table, since the table of values is really just the x and y pairs.
(We used the middle column simply to help us get the y values)

 X Value Y Value 1 7 2 12 3 17 4 22
##### Problem 2

Create a table of values of the equation y = −6x + 2.

Create the table and choose a set of x values.

 X Value Equation Y value y = −6x + 2 1 2 3 4

Substitute each x value (left side column) into the equation.

 X Value Equation Y value y = −6x + 2 1 y = −6(1) + 2 2 y = −6(2) + 2 3 y = −6(3) + 2 4 y = −6(4) + 2

Evaluate the equation (middle column) to arrive at the y value.

 X Value Equation Y value y = −6x + 2 1 y = −6(1) + 2 -4 2 y = −6(2) + 2 -10 3 y = −6(3) + 2 -16 4 y = −6(4) + 2 -22

An Optional step, if you want, you can omit the middle column from your table, since the table of values is really just the x and y pairs .(We used the middle column simply to help us get the y values)

 X Value Y value 1 -4 2 -10 3 -16 4 -22
##### Problem 3

Create a table of values of the equation y = −6x − 4

Create the table and choose a set of x values

 X Value Equation Y value y = −6x − 4 1 2 3 4

Substitute each x value (left side column) into the equation.

 X Value Equation Y value 1 y = −6(1) − 4 2 y = −6(2) − 4 3 y = −6(3) − 4 4 y = −6(4) − 4

Evaluate the equation (middle column) to arrive at the y value.

 X Value Equation Y value 1 y = −6(1) − 4 -10 2 y = −6(2) − 4 -16 3 y = −6(3) − 4 -22 4 y = −6(4) − 4 -28

An Optional step, if you want, you can omit the middle column from your table, since the table of values is really just the x and y pairs. (We used the middle column simply to help us get the y values)

 X Value Y value 1 -10 2 -16 3 -22 4 -28

## Part II. Writing Equation from Table of Values

Often, students are asked to write the equation of a line from a table of values. To solve this kind of problem, simply chose any 2 points on the table and follow the normal steps for writing the equation of a line from 2 points.

##### Problem 4

Write the equation of a line from the table of values below.

 X Value Y value 0 3 1 7 2 11 3 15

Choose any two x, y pairs from the table and calculate the slope. Since, I like to work with easy, small numbers I chose (0, 3) and (1, 7).

 X Value Y value 0 3 1 7 2 11 3 15

Substitute slope into the slope intercept form of a line.

y = mx + b
y = 4x + b

Find the value of 'b' in the slope intercept equation.

y = mx + b
y = 4x + b

Since our table gave us the point (0, 3) we know that 'b' is 3. Remember 'b' is the y-intercept which, luckily, was supplied to us in the table.

Answer: y = 4x + 3

If you'd like, you could check your answer by substituting the values from the table into your equation. Each and every x, y pair from the table should work with your answer.

##### Problem 5

Write the equation from the table of values provided below.

 X Value Y value 2 8 4 9 6 10

Choose any two x, y pairs from the table and calculate the slope. I chose (2, 8) and (4, 9).

 X Y 2 8 4 9 6 10

Substitute slope into the slope intercept form of a line.

y = mx + b
y = ½x + b

Find the value of 'b' in the slope intercept equation.

Now that we know the value of b, we can substitute it into our equation.

Answer: y = ½x + 7

If you'd like, you could check your answer by substituting the values from the table into your equation. Each and every x, y pair from the table should work with your answer.

##### Problem 6

Write the equation from the table of values provided below.

 X Value Y value 3 2 6 0 9 -2

Choose any two x, y pairs from the table and calculate the slope. I chose (2, 8) and (4, 9).

 X Y 3 2 6 0 9 -2

Substitute slope into the slope intercept form of a line.

y = mx + b
y = x + b

Find the value of 'b' in the slope intercept equation.

Now that we know the value of b, we can substitute it into our equation.

Answer: y = x + 4

If you'd like, you could check your answer by substituting the values from the table into your equation. Each and every x, y pair from the table should work with your answer.

Challenge Problem

Why can you not write the equation of a line from the table of values below?

The reason that this table could not represent the equation of a line is because the slope is inconsistent. For instance the slope of the 2 points at the top of the table (0, 1) and (1, 3) is different from the slope at the bottom (2, 8) and (3, 11).