Finding an Equation from a Table of Numbers

Mathematical equations are essentially relationships. A line equation describes the relationship between ​x​ and ​y​ values found on a coordinate plane. The equation of a line is written as ​y=mx​+​b​, where the constant ​m​ is the slope of the line, and the ​b​ is the ​y​-intercept. One of the common algebraic problem questions asked is how to find the line equation from a set of values, such as a table of numbers that correspond to the coordinates of points. Here how to solve this algebraic challenge.

Understand the Values in the Table

The numbers in a table are often the ​x​ and ​y​ values that are true for the line, which means the ​x​ and ​y​ values correspond to the coordinates of points on the line. Given that a line equation is ​y=mx​+​b​, the ​x​ and ​y​ values are numbers that can be used to arrive at the unknowns, such as the slope and the y-intercept.

Find the Slope

The slope of a line – represented by ​m​ – measures its steepness. Also, the slope gives clues to the direction of the line in a coordinate plane. The slope is constant in a line, which explains why its value can be calculated. The slope can be determined from the ​x​ and ​y​ values provided in a given table. Remember that the ​x​ and ​y​ values correspond to points on the line. In turn, calculating a line equation’s slope requires the use of two points, such as point A (​x​1, ​y​1) and point B (​x​2, ​y​2). The equation to find the slope is

(m = frac{y_2-y_1}{x_2-x_1})

to solve for the term ​m​. Notice from this equation that the slope represents the change in ​y​-value per unit of change in the ​x​-value. Let’s take the example of the first point, A, being (2, 5) and the second point, B, being (7, 30). The equation to solve for the slope then becomes

(m = frac{30-5}{7-2} = frac{25}{5} = 5)

Determine the Point Where the Line Crosses the Vertical Axis

After solving for the slope, the next unknown to solve for is the term ​b​, which is the ​y​-intercept. The ​y​-intercept is defined as the value where the line crosses the ​y​-axis of the graph. To arrive at the ​y​-intercept of a linear equation with a known slope, substitute in the ​x​ and ​y​ values from the table. Since the previous step above showed the slope to be 5, substitute the values of point A (2, 5) into the line equation to find the value of ​b​. Thus, ​y=mx​+​b​ becomes

(5=(5 × 2)+b =10+b)

so that the value of ​b​ is −5.

Check Your Work

In mathematics, it is always advisable to check your work. When the table provides other points with values for their ​x​- and ​y​-coordinates, substitute them into the line equation to verify that the value of the ​y​-intercept, or ​b,​ is correct. When you plug in the values of point B (7, 30) into the line equation, ​y​ = ​mx​ + ​b​ becomes

(30 = (5 × 7) + (-5))

Simplifying that further brings about 30 = 35 − 5, which checks out as correct. In other words, the line equation has been solved to be ​y​ = 5​x​ − 5, since the slope has been determined to be 5, and the ​y​-intercept has been determined to be −5, all from the use of the values provided by a given table of number values.