Download File - Mrs. Malinda Young, M.Ed

Graphing Systems
Of Equations
Algebra 1
Glencoe McGraw-Hill
JoAnn Evans
Two or more linear equations with the
same variables form a linear system.
There are different methods available to
solve systems of equations. Today you’ll
solve systems by graphing………but first
you need to know how to recognize a
solution when you see one.
A solution of a linear system in two variables is
an ordered pair that makes each equation a true
statement.
y
x
•
The ordered pair
solution will appear as the
point of intersection
on the graph.
The point (-1, -3) is the
solution to this system of
equations. It is a point
common to both lines.
To algebraically check the solution for a system of
equations, substitute the given values for x and y into
each equation. If a true statement is found in both
cases, then the given point IS a solution for the system.
Check if (0, 4) is a
solution of this system:
 4x  3y  12
3x  4 y  5
-4x + 3y = 12
3x - 4y = 5
-4(0) + 3(4) = 12
3(0) – 4(4) = 5
0 + 12 = 12 True
(0, 4) is NOT a solution to the
system. It satisfied only one
of the two equations.
0 – 16 ≠ 5 False
 4x  3y  12
As we found, the
point (0, 4) was a
solution for this
equation. The point
lies on the line.
3x  4y  5
The point (0, 4) was
not a solution for this
equation. The point
doesn’t on the line.
•
The solution for this system
of equations is the point of
intersection for the two
lines. In another lesson
you’ll learn how to
algebraically find the
solution to a system of
equations.
Check if (2, -1) is a
solution of this system:
y  x  1
3x  3y  9
y  x  1
( 1)  (2)  1
True
 1  1
3x  3y  9
3(2)  3( 1)  9
•
6  ( 3)  9
9  9 True
(2, -1) is a solution to the system. It satisfied both of the
equations and is the point of intersection for the lines.
Graph the system of equations. Determine the solution.
2x  y  8
 x  2 y  1
2x  y  8
 2x
2x

 y  2x  8
y  2x  8
 x  2 y  1
2y  x  1
1
1
y x
2
2
Check the point (5, 2) in both of
the original equations to verify
that it’s the solution.
If the lines you graph
aren’t precisely and
carefully drawn, it will
be very difficult to
determine the solution
from your graph.
Take time to line up your
straight edge before you
draw the lines!
A system of two linear equations can also have
no solution or infinitely many solutions.
How would a linear
system look if it had
no solution?
How would one look if
it had infinitely many
solutions?
y
y
x
x
parallel lines
same line
Graph the system of equations.
Then determine whether the
system has one solution, no
solution, or infinitely many
solutions. If there is one solution,
name the solution.
x  2y  4
1
y  x1
2
x  2y  4
 2y   x  4
2
2 2
1
y  x2
2
The lines are parallel; they
have the same slope and
different y-intercepts.
Parallel lines will never
intersect, so there is no
solution to the system.
One solution? No
solution? Infinitely many
solutions? If there is one
solution, name the
solution.
2x  y  3
8x  4 y  12
2x  y  3
 y  2x  3
y  2x  3
8x  4 y  12
 4 y  8x  12
4
4 4
y  2x  3
There are infinitely many solutions. When the two
equations are ready to graph in slope-intercept form,
it’s clear that they’re the same line.
One solution? No
solution? Infinitely many
solutions? If there is one
solution, name the
solution.
y  3
1
y  x5
3
(6, -3) appears to be the solution for the system.
Check the solution in both equations.
Bob’s not good at saving money. Today he has $220 in his
piggy bank, but every day he takes out $5 for spending
money. Ray is better at saving money. Today he has $40 in
his piggy bank and plans to add $10 every day to save for a
rainy day. In how many days will Bob and Ray’s banks have
the same amount of money in them?
Let x = # of days until they have the same amount
Let y = money in the bank
Write two equations in slope-intercept form,
one for Bob and one for Ray.
y  5x  220
Bob
y  10x  40
Ray
$
220
200
180
160
140
120
100
80
60
40
20
Check the
solution in
each
equation to
be sure.
(12, 160)
After 12 weeks,
they will each
have $160 in
the bank.
1
2 3 4
5 6 7 8 9 10 11 12 13 14
# of days
The slope of Bob’s line is -5.
Every 4 weeks, he has
$20 less.
The slope of Ray’s line is 10.
Every 2 weeks, he has
$20 more.