One common problem with linear functions is where the point of intersection of two or more linear functions is required when you express the equation graphically. Consider these two equations: 2x -- y= 10 and x + y = -1. To solve their point of intersection requires that you work them out as simultaneous equations. This method leads you to replace the value of y from the first equation (2x - 10) in the second equation, which gives x + 2x -- 10= -1, which is in turn simplified to x= 3. The final step then would be plugging in the value of x in either equation to solve for y. Replace the obtained value of x in the second equation, 3 + y= -1, and then simplify it to y= -4. The point of intersection of the two functions is (3,-4).
Another easy problem would be one that requires you to simplify the linear equation. For example, to simplify the equation 2x + 1 + 4y= -6y -4 + 5x , group together all the like terms with those with y, preferably to the left of the equal sign, and those with an x or without a variable to the right. The equation then appears in this form: 6y + 4y= 5x -2x -4 -1, which is now easier to simplify by adding the like terms together: 10y= 3x -5.
One classic problem involving linear functions is when there are two points on a graph where the line of a linear function passes through and the gradient is required. If the function is provided, the easiest way to solve for the gradient is to simplify the equation into the form y= mx + c, where m is a constant denoting the gradient of the line, and c is the y intercept, the point at which the line cuts the y axis.
Parallel lines on a Cartesian plane have the same gradient, while lines that are perpendicular to each other have gradients whose product is -1. If you have a problem where you need to differentiate between functions, comparing their gradients after simplifying them to the form y= mx + c determines whether the lines are parallel or perpendicular to each other (or neither).