Systems of Linear Equations and Their Solutions

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Systems of Linear Equations and Their Solutions
We have seen that all equations in the form Ax
By C are straight lines when graphed. Two such
equations, such as those listed below, are called
a system of linear equations. A solution to a
system of linear equations is an ordered pair
that satisfies all equations in the system. For
example, (3, 4) satisfies the system x y
7 (3 4 is, indeed, 7.) x y -1 (3 4 is
indeed, -1.)   Thus, (3, 4) satisfies both
equations and is a solution of the system. The
solution can be described by saying that x 3
and y 4. The solution can also be described
using set notation. The solution set to the
system is (3, 4) - that is, the set consisting
of the ordered pair (3, 4).
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Example Determining Whether an Ordered Pair Is
a Solution of a System
Determine whether (4, -1) is a solution of the
system x 2y 2 x 2y 6.
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The Number of Solutions to a System of Two Linear
Equations
The number of solutions to a system of two linear
equations in two variables is given by one of the
following. Number of Solutions What This
Means Graphically Exactly one ordered-pair
solution The two lines intersect at one
point. No solution The two lines are
parallel. Infinitely many solutions The two
lines are identical.
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Determining Types of solutions

• One way to determine the type of solution you
  expect to get is by looking at the coefficients
  of each variable in the two equations. Consider
  the general systems
• Compare the corresponding coefficients
• Same line
• Parallel lines
• Unique Solution

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Solving Linear Systems by Substitution

1. Solve either of the equations for one variable in
   terms of the other. (If one of the equations is
   already in this form, you can skip this step.)
2. Substitute the expression found in step 1 into
   the other equation. This will result in an
   equation in one variable.
3. Solve the equation obtained in step 2.
4. Back-substitute the value found in step 3 into
   the equation from step 1. Simplify and find the
   value of the remaining variable.
5. Check the proposed solution in both of the
   system's given equations.

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Example Solving a System by Substitution
Solve by the substitution method 5x 4y 9 x
2y -3.
Solution Step 1 Solve either of the equations
for one variable in terms of the other. We begin
by isolating one of the variables in either of
the equations. By solving for x in the second
equation, which has a coefficient of 1, we can
avoid fractions. x - 2y -3 This is the second
equation in the given system. x 2y - 3
Solve for x by adding 2y to both sides.
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Example Solving a System by Substitution
Solve by the substitution method 5x 4y 9 x
2y -3.
Solution This gives us an equation in one
variable, namely 5(2y - 3) - 4y 9. The
variable x has been eliminated.
Step 3 Solve the resulting equation containing
one variable. 5(2y 3) 4y 9 This is the
equation containing one variable. 10y 15 4y
9 Apply the distributive property. 6y 15
9 Combine like terms. 6y 24 Add 15 to both
sides. y 4 Divide both sides by 6.
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Example Solving a System by Substitution
Solve by the substitution method 5x 4y 9 x
2y -3.
Solution
Step 4 Back-substitute the obtained value into
the equation from step 1. Now that we have the
y-coordinate of the solution, we back-substitute
4 for y in the equation x 2y 3. x 2y
3 Use the equation obtained in step 1. x 2 (4)
3 Substitute 4 for y. x 8 3 Multiply. x
5 Subtract. With x 5 and y 4, the proposed
solution is (5, 4).
Step 5 Check. Take a moment to show that (5, 4)
satisfies both given equations. The solution set
is (5, 4).
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Solving Linear Systems by Addition

1. If necessary, rewrite both equations in the form
   Ax By C.
2. If necessary, multiply either equation or both
   equations by appropriate nonzero numbers so
   that the sum of the x-coefficients or the sum of
   the y-coefficients is 0.
3. Add the equations in step 2. The sum is an
   equation in one variable.
4. Solve the equation from step 3.
5. Back-substitute the value obtained in step 4 into
   either of the given equations and solve for the
   other variable.
6. Check the solution in both of the original
   equations.

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Example Solving a System by the Addition Method
Solve by the addition method 2x 7y - 17 5y
17 - 3x.
Step 2 If necessary, multiply either equation
or both equations by appropriate numbers so that
the sum of the x-coefficients or the sum of the
y-coefficients is 0. We can eliminate x or y.
Let's eliminate x by multiplying the first
equation by 3 and the second equation by -2.
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Solution
2x 7y -17
3x 5y 17
32x 37y 3(-17)
-23x (-2)5y -2(17)
6x 21y -51
-6x 10y -34
Steps 3 and 4 Add the equations and solve for the
remaining variable.
6x 21y -51
-6x 10y -34
Add
-31y -85
-31y -85
-31 -31
Divide both sides by -31.
Simplify.
y 85/31
Step 5 Back-substitute and find the value for
the other variable. Back-substitution of 85/31
for y into either of the given equations results
in cumbersome arithmetic. Instead, let's use the
addition method on the given system in the form
Ax By C to find the value for x. Thus, we
eliminate y by multiplying the first equation by
5 and the second equation by 7.
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Solution
2x 7y -17
3x 5y 17
52x 57y 5(-17)
73x 75y 7(17)
10x 35y -85
21x 35y 119
Add
31x 34
x 34/31
Step 6 Check. For this system, a calculator is
helpful in showing the solution (34/31, 85/31)
satisfies both equations. Consequently, the
solution set is (34/31, 85/31).
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Examples

• Determine the type of solution, then solve.