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

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.

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)}.

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.

x - 2y = -3 This is the second equation in the given system.
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.

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.

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.

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.

Solution

Steps 3 and 4 Add the equations and solve for the remaining variable.
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.

Solution

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

Exactly one solution	No Solution (parallel lines)	Infinitely many solutions (lines coincide)

Example

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