A system of linear equations can be solved by graphing if there are only two unknowns. If the letters used for the unknowns are not and
, we can replace them for the purpose of graphing, or simply relabel the axes to suit.
Locating a Unique Solution
If the lines intersect then the coordinates of intersection satisfy both equations.
The of the intersection is the solution of the system. For example, the point
lies on
and lies on the line
.
In this case we say the solution is unique. There is only one point of intersection that lies on two non-parallel straight lines.
Determine that there is no solution
If two lines are parallel they won’t intersect. There is no solution.
For example on the lines and
are parallel. The slope of both lines is 2. There is no
that lies on both lines. Therefore there is no
that satisfies both equations. In this case we state “no solution” to indicate we have arrived at that conclusion. This is a different conclusion than leaving a blank or writing “idk”.
Infinitely many solutions
If the lines are identical we call them coincident. There are infinitely many solutions.
For example, the lines and
are just different arrangements of the same relation. When graphed the lines are identical. In this case every point that lies on
lies on
. In this case, there are infinitely many solutions.
To learn more check out Khan Academy: Introduction to systems of equations
Draw graphs to find the common point
In this practice applet there is a unique solution to each pair of equations.
Position the points given on each one of the two lines: first the blue, then the green. Use the intersection to determine the value of and of
that satisfies both equations.
printable resource: Grade 10 Unit 7 Solve a system by graphing