Solving Linear Equation using graphs

By looking at where lines intersect (meet) on a graph, the solution of the two equations can be found. Put both equations into slope-intercept form (y = mx +b). Graph both equations on the same Cartesian plane using a straightedge. Graph the second equation in the same manner. Extend the two lines until you see where they intersect. Check your solution by plugging the point into each equation

Systems with no solution and infinite solutions

When you are trying to calculate the solution of a system of linear equations, you can will arrive at one of three distinct cases:

These cases only apply to systems of two lines. If you are working with systems with three or more linear equations (lines), you cannot use the blanket generalizations made below.

• The system has exactly 1 solution

Systems have 1 and only 1 solution when the two lines have different slope. Think about it, if the two lines have different slopes then eventually at some point they must meet. After all the lines are not parallel.

• system has no solutions

Systems have no solution when the lines are parallel (ie have the same slope) and the lines have different y-intercepts.

As an example look at the following two lines

Line 1: y = 5x +13

Line 2: y = 5x + 12

• The system has infinite solutions

Systems have infinite solutions when the lines are parallel and the lines have the same y-intercept. If two lines have the same slope (ie are parallel) and the same y-intercept, they are actually the same exact line. In other words, systems have infinite solutions when the two lines are the same line.

It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point). In general, a system of linear equations may behave differently than expected if the equations are linearly dependent, or if two or more of the equations are inconsistent.