You’ve seen how a line in a coordinate plane may be represented by an equation.
The coordinates of the points on the line are the x- and y-values that satisfy the equation. The points are the solutions to the equation of the line.
You also know that two lines in the same plane are either parallel or intersect at a point.
Since both the lines can be represented by equations, the equations can be used to find the coordinates of the point of intersection of the lines.
The point of intersection is on both lines, so it is a solution to both equations.
When a two or more equations with two or more variables are considered together so that simultaneous solutions to all of them are sought, the equations are said to be a system of equations.
Here, there are two linear equations with two variables, called a system of two equations in two unknowns, or a 2-by-2 system for short.
The coordinates of the point of intersection depends entirely on the parameters appearing in the lines’ equations.
The simulation allows you to experiment with 2-by-2 systems.
The point of intersection does not depend on the form of the equation of the line, so you will be able to enter the equation in either the slope-intercept form or the standard form.
Choose the form you prefer by using the drop-down list, then enter the coefficients for the form you have chosen.
The coefficients for the slop-intercept form are displayed as fractions, but they should be entered as decimals. They will change back to fractions after you press enter.
You can also define a line by entering a pair of points. Choose “Points form” from the drop-down list and enter the coordinates of the point.
You can modify the lines by clicking and dragging. Dragging any point except the point of intersection will pivot the line about the other point on the line.
Dragging a line will move the line as a whole, and the line will retain its slope while being dragged. As you modify the lines, the parameters defining the line will change, as will the point of intersection.
As the lines are modified, their point of intersection, if any, will be updated as an ordered pair, along with a classification of the system. A system is said to be consistent if it has a solution. If it has no solution, it is inconsistent.
Try as many different cases as you can think of. What is the solution when the lines are parallel? Can a system have more than one solution? What happens if you enter the same line with two different methods or equation forms?