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Systems of Linear
Equations in Two
Variables
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 above, 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).
Text Example
Determine whether (4, -1) is a solution of the system
x + 2y = 2
x – 2y = 6.
Solution Because 4 is the x-coordinate and -1 is the y-coordinate of (4, -1),
we replace x by 4 and y by -1.
x + 2y = 2
x – 2y = 6
?
?
4 + 2(-1) = 2
4 – 2(-1) = 6
?
?
4 + (-2) = 2
4 – (-2) = 6
?
2 = 2 true
4+2=6
6 = 6 true
The pair (4, -1) satisfies both equations: It makes each equation true. Thus, the
pair is a solution of the system. The solution set to the system is {(4, -1)}.
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Solving Linear Systems by Substitution
• 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.)
• Substitute the expression found in step 1 into the
other equation. This will result in an equation in
one variable.
• Solve the equation obtained in step 2.
• Back-substitute the value found in step 3 into the
equation from step 1. Simplify and find the value
of the remaining variable.
• Check the proposed solution in both of the
system's given equations.
Text Example
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.
This is the second equation in the given system.
x - 2y = -3
x = 2y - 3
Solve for x by adding 2y to both sides.
Step 2 Substitute the expression from step 1 into the other equation. We
substitute 2y - 3 for x in the first equation.
x = 2y – 3
5 x – 4y = 9
Text Example cont.
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.
This is the equation containing one variable.
5(2y – 3) – 4y = 9
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.
2
Text Example cont.
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.
Use the equation obtained in step 1.
x = 2y – 3
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)}.
Solving Linear Systems by Addition
• If necessary, rewrite both equations in the form Ax +
By = C.
• 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 ycoefficients is 0.
• Add the equations in step 2. The sum is an equation in
one variable.
• Solve the equation from step 3.
• Back- substitute the value obtained in step 4 into either
of the given equations and solve for the other variable.
• Check the solution in both of the original equations.
Text Example
Solve by the addition method:
2x = 7y - 17
5y = 17 - 3x.
Solution
Step 1 Rewrite both equations in the form Ax + By = C. We first arrange
the system so that variable terms appear on the left and constants appear on
the right. We obtain
2x - 7y = -17
3x + 5y = 17
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.
3
Text Example cont.
Solution
2x – 7y = -17
3x + 5y = 17
Steps 3 and 4
Multiply by 3.
Multiply by -2.
3•2x – 3•7y = 3(-17)
-2•3x + (-2)5y = -2(17)
6x – 21y = -51
-6x + 10y = -34
Add the equations and solve for the remaining variable.
6x – 21y = -51
-6x – 10y = -34
-31y = -85
Add:
-31y = -85
-31
-31
y
= 85/31
Divide both sides by -31.
Simplify.
Step 5 Back-substitute and find the value for the other variable. Backsubstitution 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.
Text Example cont.
Solution
2x – 7y = -17
3x + 5y = 17
Multiply by 5.
Multiply by 7.
5•2x – 5•7y = 5(-17)
7•3x + 7•5y = 7(17)
10x – 35y = -85
21x + 35y = 119
Add: 31x
x
= 34
= 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)}.
The Number of Solutions to a
System of Two Linear Equations
The
The number
number of
of solutions
solutions to
to aa system
system of
of two
two linear
linear equations
equations in
in two
two variables
variables
is
is given
given by
by one
one of
of the
the following.
following.
Number
Number of
of Solutions
Solutions
Exactly
Exactly one
one ordered-pair
ordered-pair solution
solution
No
No solution
solution
Infinitely
Infinitely many
many solutions
solutions
y
y
x
Exactly one solution
What
What This
This Means
Means Graphically
Graphically
The
The two
two lines
lines intersect
intersect at
at one
one point.
point.
The
The two
two lines
lines are
are parallel.
parallel.
The
The two
two lines
lines are
are identical.
identical.
y
x
No Solution (parallel lines)
x
Infinitely many solutions
(lines coincide)
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Example
Solve the system
=4
2x + 3y
-4x - 6y = -1
Solution:
2 (2x + 3y = 4)
multiply the first equation by 2
-4x - 6y = -1
4x + 6y = 8
-4x - 6y = -1
0=7
Add the two equations
No solution
Systems of Linear
Equations in Two
Variables
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