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SOLVING SYSTEMS OF LINEAR EQUATIONS BY THE SUBSTITUTION METHOD

SOLVING SYSTEMS OF LINEAR EQUATIONS BY THE SUBSTITUTION METHOD

Definitions:

• System of linear equations : consists of two or more linear equations with the same variables.

• Consistent: The system is consistent if there is exactly one solution.

• Inconsistent: The system is inconsistent if there is no solution. This happens when the two equa-
tions represent parallel lines .

• Dependent: The system is dependent if there is an infinite number of ordered pairs as solutions.
This occurs when the two equations represent the same line.

Steps for the Substitution Method:

1. Choose one of the equations and solve for one variable in terms of the other variable.

2. Substitute the expression from Step 1 into the other equation.

3. Solve the equation from Step 2. (There will be one equation with one variable).

4. Substitute the solution from Step 3 into either of the original equations. This will give the value of
the other
variable.

Important Properties :

• The Substitution Method is useful when one equation can be solved very quickly for one of the
variables.

• If the equation in Step 3 above is a false statement (such as 7 = 2), then the system is inconsistent.

• If the equation in Step 3 above is a true statement (such as 0 = 0), then the system is dependent.

Common Mistakes to Avoid:

• Remember that a system of linear equations is not completely solved until values for both x and y
are found. To avoid this mistake, write all answers as an ordered pair .

• Remember that all ordered pairs are stated with the x-variable first and the y-variable second,
namely, (x, y).

• If the first equation is used to solve for the variable, substitute it into the second equation. Otherwise,
this will incorrectly lead to the statement 0 = 0.

PROBLEMS

1. Solve

2x + y = 5
3x + 2y = -8

Notice that the first equation can be solved
easily for y, giving us

2x + y = 5
y = -2x + 5

This is what we will now substitute into the
y variable in our second equation. This gives
us:

3x + 2(-2x + 5) = -8
3x - 4x + 10 = -8
-x + 10 = -8
-x = -18
x = 18

Next, we need to find the value of our y
variable by substituting x = 18 into one of
the equations. Since we already know that
y = -2x + 5, substituting in this equation
gives us:
y = -2(18) + 5
y = -36 + 5
y = -31
Answer: (18,-31)

3. Solve

x - y = -3
4x + 3y = -5

Notice that the first equation can be solved
quickly for either x or y. We will solve for x.

x - y = -3
x = y - 3

We now substitute this into the x variable in
our second equation.

4(y - 3) + 3y = -5
4y - 12 + 3y = -5
7y - 12 = -5
7y = 7
y = 1

We now substitute y = 1 into one of our
equations in order to find the value of x.
Since we already know that x = y - 3, sub-
stituting y = 1 into this equation yields

x = 1 - 3
x = -2

Answer: (-2, 1)

5. Solve
2x + 3y = 5
x - 4y = 6

Notice that the second equation can be
solved easily for x.

x - 4y = 6
x = 4y + 6

We will now substitute this into the x vari-
able in our first equation.

2(4y + 6) + 3y = 5
8y + 12 + 3y = 5
11y + 12 = 5
11y = -7


Finally, we need to solve for the x variable by
substituting into one of our equa-
tions. Since we already know that x = 4y+6
substituting into this equation yields

Answer:
2. Solve

4x + 3y = 10
2x + y = 4

Notice that we can quickly solve for y using
the second equation.

2x + y = 4
y = -2x + 4

We will now substitute this into the y vari-
able in our first equation.

4x + 3(-2x + 4) = 10
4x - 6x + 12 = 10
-2x + 12 = 10
-2x = -2
x = 1

We now need to find the value of y by sub-
stituting x = 1 into one of our equations.
Since we already have that y = -2x + 4,
substituting into this equation gives

y = -2(1) + 4
y = -2 + 4
y = 2
 
Answer: (1, 2)

4. Solve

2x - y = 3
-6x + 3y = 9

Notice that the first equation can be solved
quickly for y.

2x - y = 3
-y = -2x + 3
y = 2x - 3

We now substitute this into the y variable in
our second equation.

-6x + 3(2x - 3) = 9
-6x + 6x - 9 = 9
-9 = 9

Since this is a false statement, the system is
inconsistent. Therefore, there is no solution.

Answer: No Solution










 

6. Solve

4x + y = 10
3x + 2y = 5

Notice that the first equation can be easily
solve for y.

4x + y = 10
y = -4x + 10

We then substitute this into the y variable
in the second equation.

3x + 2(-4x + 10) = 5
3x - 8x + 20 = 5
-5x + 20 = 5
-5x = -25
x = -5

Finally, we need to find the value of y by sub-
stituting x = -5 into one of our equations.
Since we already know that y = -4x + 10,
substituting into this equation gives us

y = -4(-5) + 10
y = 20 + 10
y = 30

Answer: (-5, 30)

 

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