Graphing Systems of Linear Equations When you are looking at the graph of a linear
equation you are looking at all the ordered
pairs that satisfy its equation.
Suppose you have a graph of two lines (look
to the right). Can you see any points that
satisfy both of the lines’ equations?
These points are called intersection points
and they are points satisfying both lines!
If you are asked to find the solution to a
System (more than 1 equation), you are trying
to find a point or points satisfying both lines.
A solution to a system of linear equations
is the _____________of the two lines.
:
The lines being graphed above are:
1.
When graphing a system of equations, there
are 3 things that can happen.
1) The lines have 1 intersection. In
this case only 1 point satisfies both
lines so the system has _____
solutions.
2) The lines are parallel and don’t have
any intersections. In this case the
system has _____solutions
3) The lines look the same when you
graph them. So system has
_______________solutions.
2.
Hint: Use the following form of an equation.
y = (Slope) x + y-intercept
What is the solution to your system of
equations? ____________
Check your answer. Remember a solution
satisfies both equations. So
(
) = -2 (
(
)=3(
)
) +
4
-
6
and
Find the solutions of each system of linear equations by graphing. Determine whether each
system has one solution, no solutions, or infinitely many solutions. If the system has only one
solution, write the solution as an ordered pair. To help graph, write your lines in the form
y = (slope) x + y-intercept.
1.
𝑥 + 𝑦 = −1
𝑥−𝑦 =3
Does this system of equations have one solution, no solutions, or infinitely many solutions?
_________________________
If there is only one solution give the solution: _______________
The equations in slope-intercept form will look like:
𝒙 + 𝒚 = −𝟏 → 𝒚 = −𝒙 − 𝟏
Slope: ____
y-intercept: _____
𝒙 − 𝒚 = 𝟑 → 𝒚 = 𝒙 − 𝟑
Slope: ____
y-intercept: _____
Note: You can tell a lot about your solutions when you know the equations slopes and yintercepts. For instance if the slopes are different the lines must eventually cross
somewhere. So you will have 1 solution.
2.
3𝑥 + 2𝑦 = 6
6𝑥 + 4𝑦 = 24
Does this system of equations have one solution, no solutions, or infinitely many solutions?
_________________________
If there is only one solution give the solution: _______________
3.
𝑥 − 𝑦 = −7
𝑥 = −3
Does this system of equations have one solution, no solutions, or infinitely many solutions?
_________________________
If there is only one solution give the solution: ______________
4.
3𝑥 + 𝑦 = 9
6𝑥 + 2𝑦 = 18
Does this system of equations have one solution, no solutions, or infinitely many solutions?
_________________________
If there is only one solution give the solution: _______________
On example #2 and #4, the slopes of our lines are the same telling us the lines are parallel.
However; on #2 you have no solution and on #4 you have infinitely many solutions. What
determines the difference in these 2 cases? Can you form a conclusion about what types of
solutions you have based on the way the equations look- either in the form they are given or slopeintercept form?