Algebraic linear equations are mathematical functions that, when graphed on a Cartesian coordinate plane, produce x and y values in the pattern of a straight line. The standard form of the linear equation can be derived from the graph or from given values. Linear equations are fundamental to algebra, and thus fundamental to all higher mathematics.
Note that the standard form of a linear equation is:
y = mx + b
Where m = slope and b = y-intercept.
Calculate the slope of the line. The slope can be found by selecting two points on the line, determining the vertical rise and the horizontal run between the points and dividing them. For example, if (3,4) and (5,6) are on the line, the slope between them would be (5 - 3) / (6 - 4), simplified to (2) / (2), simplified to 1. Include negative values, since slopes can be positive or negative.
Determine or calculate the y-intercept of the line. The y-intercept is the y-coordinate of the point where the line passes through the y-axis of the coordinate plane. For example, if the point of intersection with the y-axis is (0,5), the y-intercept would be 5. The y-intercept can be found by physically locating it on the graph or by locating the given point on the line that has an x-coordinate of 0. That point is the point of intersection. The y-intercept will be positive if it intersects the y-axis above the x-axis or negative if it intersects below the x-axis.
Write the equation y = mx + b, substituting the values for m and b you calculated or determined. The m will be your slope, and the b will be your y-intercept. Leave the y and x variables in the equation as letter variables. Include the sign of the numbers you plug in. For example, if I discovered my slope to be -3 and my y-intercept to be 5, my linear equation would be y = -3x + 5. The linear equation is complete and correctly written when the (m) and (b) are properly incorporated into the equation.
Factor negative signs into the linear equation carefully. If b = -8 and m = 5, the algebraic linear equation would be written y = 5x + (- 8), or simplified, y = 5x - 8.
When in doubt, check your work.