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 (x1, y1) and point B (x2, y2). The equation to find the slope is (y1-y2)/(x1-x2) 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 (30-5)/(7-2), which simplifies to (25)/(5), or a slope of 5.
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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, which is simplified into 5=(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=5x-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.