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.

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.

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

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

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

so that the value of ​b​ is −5.

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

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.