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

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

## 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

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

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.