# Which Table Represents a Linear Function

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

### Equations as Relationships

The equation of a line expresses a relationship betwixt x and y values on the coordinate airplane. For instance, the equation
$$y = x$$
expresses a human 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 10 value.

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

Since, every bit we simply wrote, every linear equation is a relationship of ten and y values, nosotros tin create a tabular array of values for any line. These are simply the
$$x$$

and
$$y$$

values that are
true
for the given line. In other words, a table of values is just some of the points that are on the line.

##### Example i

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

Table of Values

 $$\blue x \text { value}$$ Equation $$\red y \text{ value}$$ y = 10 + 1 $$\blueish three$$ $$y = ( \blue iii ) + one$$ $$\red 4$$ $$\blue four$$ y = ($$\blue iv$$ ) + ane $$\red 5$$ $$\blue five$$ $$y = (\blue v ) + 1$$ $$\red six$$ $$\blueish six$$ $$y = ( \blue 6) + one$$ $$\red seven$$
##### Example two

Equation:

y = 3x + 2

Table of Values

 Ten Value Equation Y value y = 3x + 2 one y = 3(1) + 2 5 2 y = 3(2) + 2 7 3 y = 3(iii) + two 11 4 y = 3(four) + 2 fourteen

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

### Exercise Creating a Table of Values

##### Problem i
Popular:   The Phosphorus Cycle Differs From the Biogeochemical Cycles in That

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

Create the table and choose a set of x values.

 10 Value Equation Y value y = 5x + ii one 2 3 four

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

 Ten Value Equation Y value y = 5x + 2 1 y = five(1) + 2 ii y = 5(2) + 2 three y = 5(three) + two 4 y = 5(4) + ii

Evaluate the equation (center cavalcade) to get in at the y value.

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

An Optional step, if you lot want, you tin can omit the middle column from your table, since the table of values is really just the x and y pairs.

(Nosotros used the heart column simply to help us get the y values)

 Ten Value Y Value ane 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 + two 1 2 iii four

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 three y = −six(3) + two 4 y = −6(iv) + 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 -four 2 y = −6(ii) + ii -10 3 y = −6(three) + two -16 four y = −6(four) + 2 -22

An Optional step, if you want, you tin can omit the middle column from your table, since the table of values is really just the 10 and y pairs .(We used the middle column merely to assist us go the y values)

 X Value Y value one -four 2 -10 3 -16 4 -22

##### Problem 3

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

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

 X Value Equation Y value 1 y = −six(ane) − four 2 y = −6(2) − four three y = −six(three) − 4 4 y = −6(four) − iv

Evaluate the equation (heart column) to get in at the y value.

 X Value Equation Y value 1 y = −vi(1) − 4 -10 2 y = −6(2) − four -xvi 3 y = −6(3) − 4 -22 iv y = −half-dozen(4) − 4 -28

An Optional step, if yous want, you can omit the middle column from your tabular array, since the table of values is really but the x and y pairs.
(We used the centre column only to help the states get the y values)

 X Value Y value ane -10 2 -xvi three -22 four -28

## Function 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 tabular array and follow the normal steps for writing the equation of a line from two points.

##### Problem four

Cull whatsoever two x, y pairs from the table and calculate the slope. Since, I like to work with easy, small-scale numbers I chose (0, 3) and (one, 7).

 X Value Y value 3 1 7 2 11 3 15

Observe the value of ‘b’ in the slope intercept equation.

y = mx + b

y = 4x + b

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

##### Problem 5

Write the equation from the table of values provided below.

 Ten Value Y value two 8 4 ix 6 10

Find the value of ‘b’ in the gradient intercept equation.

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

Answer:
y = ½ten + 7

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

##### Trouble half-dozen

Find the value of ‘b’ in the slope intercept equation.

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

Answer:
y =
x + 4

If you’d similar, you could check your answer by substituting the values from the tabular array into your equation. Each and every 10, y pair from the tabular array should piece of work with your answer.

Challenge Problem

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

The reason that this table could
not
correspond the equation of a line is because the slope is inconsistent. For instance the gradient of the ii points at the top of the table (0, one) and (1, 3) is dissimilar from the slope at the lesser (2, eight) and (3, 11).

## Which Table Represents a Linear Function

Source: https://www.mathwarehouse.com/algebra/linear_equation/linear-equation-table-examples-graphs.php