Download 6.3 Solving Systems with Substitution

6.3 Solving Systems with Substitution
While graphing is useful for an estimate, the main way that we can solve a system to get an exact answer
is algebraically. There are a few useful methods to do this, and we will begin with the substitution method. The
general idea with this method is to isolate a single variable in one equation and substitute that into the other
equation.
Isolating a Variable
Consider the following system of equations.
3 + 1 = 1
3 + 21 = 4
It is always best to check if one variable has a coefficient of one and isolate that variable. Remember that
a coefficient is a number multiplied by a variable. That means that a coefficient of one will mean that the variable
doesn’t have a number in front of it because the one is understood to be there, and we don’t write it. In this case,
notice that the 1 in the first equation has a coefficient of one. It would probably be easiest to isolate that variable.
Let’s do so.
3 + 1 − 3 = 1 − 3
1 = 1 − 3
Substitution
Now that we know what 1 is equal to in the first equation, we can substitute that expression for 1 in the
second equation. Be careful to not plug back into the first equation or else we’ll end up with infinite solutions every
time. Since we want a solution that is true in both equations, we must use both equations.
3 + 21 = 4
3 + 2(1 − 3) = 4
3 + 2 − 6 = 4
−3 + 2 = 4
Now that we have it down to a simple two-step equation, we can solve like normal and get the following:
−3 + 2 − 2 = 4 − 2
−3
2
=
−3
−3
=−
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2
3
Finding the Second Variable Value
Now that we know what equals, we can substitute that back into either of the original equations to find
what the 1 coordinate is at the point of intersection of the two lines. It is also a good idea to plug in this value
into both equations to make sure they give the same 1 value. We’ll start with the first equation.
3 . 1 1
2
3 ÁV Â . 1 1
3
V2 . 1 1
V2 . 2 . 1 1 . 2
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Double Check
This means that the solution should be the point (− , 3, but we found that 1 value using the first
equation. We need to make sure this point is on the second line as well, so let’s substitute the values into that
equation.
3 . 21 4
2
3 ÁV Â . 23 4
3
V2 . 6 4
V , 3
That statement is true and therefore the point is on the second line as well. So our solution is the point
for this system. Just for some extra confidence, examine the following graph of the system of equations
and notice that the point we found is indeed the point of intersection.
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Coefficients Other Than One
It may be the case that we have all coefficients with values other than one. We can still use substitution,
but we’ll have to be a bit more careful isolating one variable at the beginning. Let’s consider the following system
of equations.
2 + 41 = 8
3 + 21 = 7
In this case, it might be easier to solve the first equation for because the coefficient for 1 and the 8 will
easily divide by 2. So let’s isolate the in the first equation as follows:
2 . 41 − 41 = 8 − 41
2 8 − 41
=
2
2
= 4 − 21
Now substitute that value into the second equation as follows:
3 . 21 = 7
34 − 21 . 21 = 7
12 − 61 . 21 = 7
12 − 41 = 7
12 − 12 − 41 = 7 − 12
−41 = −5
−41 −5
=
−4
−4
1=
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5
4
Now that we have the 1 coordinate, we can plug in to find the value.
2 + 41 = 8
5
2 + 4 Á Â = 8
4
2 + 5 = 8
2 + 5 − 5 = 8 − 5
2 = 3
2 3
=
2
2
=
3
2
So our solution is [ , \. We’ll leave it as an exercise to double check using the second equation.
Infinite and No Solutions
It is still possible to get infinite solutions or no solution for a system of equations. After the substitution
step, if we get down to a number equals a number statement that is always true, there are infinite solutions. If we
get down to a number equals a number statement that is false, there are no solutions. This is really the application
of what we learned earlier in this unit about solving equations with one variable and getting infinite or no solutions.
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Lesson 6.3
Solve the following systems using the substitution method.
1.
3.
5.
7.
9.
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2 + 81 = 12
− 21 = 0
1=5
2 − 1 = 9
2 . 1 = −16
− 21 = −28
+1 =2
−2 + 41 = −19
2 . 1 = 4
21 = −4 . 8
2.
4.
6.
8.
10.
+1 =7
2 . 1 = 5
1 = − + 1
2 . 31 = 6
41 = 8
2 . 51 = 11
+ 21 = 4
3 − 41 = −3
.1 =2
.1 =5
11.
13.
15.
17.
19.
1 = 3
3 + 31 = 4
− 31 = 0
.1 =2
=2
2 + 1 = 4
= 21
1 =−1
+ 21 = 0
3 . 41 = 4
12.
14.
16.
18.
20.
1 = 2 + 3
1 = 4 − 1
2 − 1 = −9
−3 . 1 = 15
4 = 31 + 3
=2
− 21 = −1
31 = + 4
21 = −6
. 21 = −1
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21.
23.
25.
27.
29.
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− 41 = 1
2 − 81 = 2
=0
3 − 61 = 12
2 − 31 = −24
.
1
= −5
+1 =6
21 = −2 . 2
2 = 61 − 14
31 − = 7
22.
24.
26.
− 21 = 3
4 − 81 = 12
= 21 − 3
= 21 . 4
− 21 = 12
= −21 − 2
28.
30.
+ 21 = 7
2 − 81 = 8
1 = − . 3
21 . 2 = 4
Write and solve a system of equations using any method (graphing, elimination, or substitution) for each of the
following situations.
31. Leonard sells small watermelons for $7 each and large watermelons for $10 each. One day the number of
small watermelons he sold was fifteen more than the number of large watermelons, and he made a total of $394.
How many small and how many large watermelons did he sell?
32. The perimeter of a rectangle is 28 cm. The length of the rectangle is 2 cm more than twice the width. Find
the dimensions of the rectangle.
33. The sum of Julian’s and Kira’s age is 58. Kira is fourteen less than twice as old as Julian. What are their ages?
34. A 3% solution of sulfuric acid was mixed with an 18% solution of sulfuric acid to produce an 8% solution.
How much 3% solution and how much 18% solution were used to produce 15 L of 8% solution?
35. Supplementary angles are two angles whose measures have the sum of 180 degrees. Angles X and Y are
supplementary, and the measure of angle X is 24 degrees greater than the measure of angle Y. Find the measures
of angles X and Y.
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36. At the end of the 2000 baseball season, the New York Yankees and the Cincinnati Reds had won a total of 31
World Series. The Yankees had won 5.2 times as many World Series as the Reds. How many World Series did
each team win?
37. Peanuts worth $2.25 a pound were mixed with cashews worth $3.25 a pound to produce a mixture worth
$2.65 a pound. How many pounds of each kind of nuts were used to produce 35 pounds of the mixture?
38. Ernesto spent a total of $64 for a pair of jeans and a shirt. The jeans cost $6 more than the shirt. What was
the cost of the jeans?
39. The perimeter of a rectangular garden is 68 feet. The length of the garden is 4 more than twice the width.
What are the dimensions of the garden?
40. The Future Teachers of America Club at Paint Branch High School is making a healthy trail mix to sell to
students during lunch. The mix will have three times the number of pounds of raisins as sunflower seeds.
Sunflower seeds cost $4.00 per pound, and raisins cost $1.50 per pound. If the group has $34.00 to spend on the
raisins and sunflower seeds, how many pounds of each should they buy?
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