Thinking
Mathematically
Systems of Linear Equations
Systems of Linear Equations and
Their Solutions
A solution to a system of linear equations is
an ordered pair that satisfies all equations in
the system.
For a system with one solution, the
coordinates of the point of intersection of
the lines is the system’s solution.
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Solving Linear Systems by Substitution
1. Solve either of the equations for one variable in
terms of the other.
2. Substitute the expression from 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.
Example Solving a System by
Substitution
Solve by the substitution method:
5x - 4y = 9
x - 2y = -3
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Solution
Step 1 Solve one 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
x = 2y - 3
Solution cont.
Step 2 Substitute the expression from step
1 into the other equation.
We substitute 2y - 3 for x in the first equation:
5(2y-3) - 4y = 9
The variable x has been eliminated.
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Solution cont.
Step 3 Solve the resulting equation containing one
variable.
5(2y - 3) - 4y = 9
10y - 15 - 4y = 9
6y - 15 = 9
6y = 24
y=4
Solution cont.
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
x = 2(4) - 3
x=8-3
x=5
With x = 5 and y = 4, the proposed solution is (5, 4).
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Solution cont.
Step 5 Check the proposed solution in both of the
system’s given equations.
(5, 4) will satisfy both given equations. The solution
set is {(5, 4)}.
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.
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Solving Linear Systems by Addition
4. Solve the equation in one variable.
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.
Solving a System by the Addition
Method
Solve by the addition method:
7x = 5 - 2y
3y = 16 - 2x.
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Solution
Step 1 If necessary, 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:
7x + 2y = 5
2x + 3y = 16
Solution cont.
Step 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.
We can eliminate x or y. Let’s eliminate y by
multiplying the first equation by 3 and the second
by -2.
7x + 2y = 5 multiply by 3
21x + 6y = 15
2x + 3y = 16 multiply by -2
-4x - 6y = -32
Step 3 Add the equations.
17x + 0 = -17
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Solution cont.
Step 4 Solve the equation in one variable. We
solve 17x = -17 by dividing both sides by 17.
17x = -17
17
17
x = -1
Step 5 Back-substitute and find the value for the
other variable. We can back-substitute -1 for x
into either one of the given equations. We’ll use
the second one.
3y = 16 - 2(-1) = 16 + 2 = 18
y=6
Solution cont.
Step 6 Check. If you substitute -1 in for x in
each equation and 6 in for y, you will see
that (-1, 6) satisfies both given equations.
The solution is (-1, 6) and the solution set is
{(-1, 6)}.
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Linear Systems Having No Solution or
Infinitely Many Solutions
If both variables are eliminated when a system
of linear equations is solved by
substitution or addition, one of the
following is true.
1. There is no solution if the resulting
statement is false.
2. There are infinitely many solutions if the
resulting statement is true.
Thinking
Mathematically
Systems of Linear Equations
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