Another way to solve systems of equations is by substitution. In this method, you solve an equation for one variable, then you substitute that solution in the other equation, and solve.
Example:
1. Problem: Solve the following system:
x + y = 11
3x - y = 5
Solution: Solve the first equation for y (you could
solve for x — it doesn't matter).
y = 11 - x
Now, substitute 11 - x for y in the second
equation. This gives the equation one variable, which
earlier algebra work has taught you how to solve for.
3x - (11 -x) = 5
3x - 11 + x = 5
4x = 16
x = 4
Now, substitute 4 for x in either equation and
solve for y. (We use the first equation below.)
4 + y = 11
y = 7
The solution is the ordered pair, (4, 7).
The last method, addition, is probably the most complicated, but
is necessary when dealing with more complex systems, such as
systems with three or more variables. The idea behind the addition
method is to replace an equation with a combination of the equations in
the system. To obtain such a combination, you multiply each equation
by a constant and add. You choose the constants so that the resulting
coefficient of one of the variables will be 0.
Example:
1. Problem: Solve the following system:
5x + 3y = 7
3x - 5y = -23
Solution: Multiply the second equation by 5 to make the
x-coefficient a multiple of 5. (This works because
it does not change the equation (see the
multiplication property).)
15x - 25y = -115
Next, multiply the first equation by -3 and add it
to the second equation. This gets rid of the x
term.
-15x - 9y = -21
15x - 25y = -115
-----------------
- 34y = -136
Now, solve the second equation for y. Then substitute
the result into the first equation and solve for x.
-34y = -136
y = 4
5x + 3(4) = 7
5x + 12 = 7
5x = -5
x = -1
The solution is the ordered pair, (-1, 4).
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Example:
1. Problem: Solve the following system:
x + y + z = 4
x - 2y - z = 1
2x - y - 2z = -1
Solution: Start out by multiplying the first equation by -1
and add it to the second equation to eliminate x
from the second equation.
-x - y - z = -4
x - 2y - z = 1
----------------
-3y - 2z = -3
Now eliminate x from the third equation by
multiplying the first equation by -2 and add it
to the third equation.
-2x - 2y - 2z = -8
2x - y - 2z = -1
------------------
-3y - 4z = -9
Next, eliminate y from the third equation by
multiplying the second equation by -1 and adding
it to the third equation.
3y + 2z = 3
-3y - 4z = -9
--------------
-2z = -6
Solve the third equation for z.
-2z = -6
z = 3
Substitute 3 for z in the second equation and
solve for y.
-3y - 2z = -3
-3y - 2(3) = -3
-3y - 6 = -3
-3y = 3
y = -1
Lastly, substitute -1 for y and 3 for
z in the first equation and solve for x.
x + (-1) + 3 = 4
x + 2 = 4
x = 2
The answer is (2, -1, 3).
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Example:
1. Problem: Graph y < x.
Solution: First graph the equation y = x.
However, the line must be drawn dashed because the less
than sign tells us the line is not included in the
solution.
Next, test a point that is located above the line and
one that is below the line. Any point you pick above
the line, such as (0, 2), y is greater
than x, so points above the line are not included
in the solution. Points below the line, such as (3, -3)
have a y-value that is less than the x-value,
so all points below the line are included in the
solution. Example Graph.
To solve a system or conjunction of inequalities, it is easiest
to graph each of the inequalities and then find their intersection.
Example:
1. Problem: Graph the following system:
2x + y >= 2
4x + 3y <= 12
(1/2) <= x <= 2
y >= 0
Solution: Example Figure.
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Take the quiz on systems of equations and inequalities. The quiz is very useful for either review or to see if you've really got the topic down.