SOLVING LINEAR EQUATIONS USING SUBSTITUTION METHOD

Solve the following systems of equations by substitution.

Example 1 :

0.2x + 0.3y  =  1.3

0.4x + 0.5y  =  2.3

Solution :

0.2 x + 0.3 y = 1.3 ------(1) 

0.4 x + 0.5 y = 2.3  ------(2) 

Multiply both (1) and (2) by 10,

2 x + 3 y = 13 ----(1)

4 x + 5 y = 23 -----(2)

Step 1 :

Find the value of one variable in terms of other variable, say y in terms of x

3y  =  13 - 2x

y  =  (13 - 2x)/3

Step 2 :

By applying the value of y in the second equation, we get

4 x + 5 [(13 - 2x)/3]  =  23

12 x + [5 (13 - 2 x)]/3  =  23

12 x + 65 - 10 x  =  69

2x  =  69 - 65

2 x  =  4

x  =  2

Step 3 :

Now, we have to apply the value of x in the equation

y  =  (13 -2x)/3

y  =  (13 -2(2))/3

y  =  (13 -4)/3

y  =  9/3

y  =  3

So, the solution is (2, 3).

Example 2 :

√2x + √3y  =  0

√3x - √8y  =  0

Solution :

Step 1 :

Find the value of one variable in terms of other variable, say y in terms of x

√3 y  =  - √2 x

y  =  - (√2/√3) x

Step 2 :

By applying the value of y in the second equation, we get

√3x - √8 [- (√2/√3) x]  =  0

√3x + (√16/√3) x)  =  0

(3x + 4x)/√3  =  0

7x/√3  =  0

7x  =  0

x  =  0

Step 3 :

Now, we have to apply the value of x in the equation

y  =  - (√2/√3) x

y  =  - (√2/√3) (0)

y  =  0

So, the solution is (0, 0).

Example 3 :

(3x/2) - (5y/3)  =  -2

(x/3) + (y/2)  =  13/6

Solution :

(3x/2) - (5y/3)  =  -2  --------(1)

(x/3) + (y/2)  =  13/6   --------(2)

We are going to take L.C.M for both equations.

(9x - 10y)/6  =  -2

9x - 10y  =  -12 ------(1)

(x/3) + (y/2)  =  13/6

(2x + 3y)/6  =  13/6

2x + 3y  =  13  ------(2)

Step 1 :

Find the value of one variable in terms of other variable, say y in terms of x

10 y  =  9x + 12  

y  =  (9x + 12)/10

Step 2 :

By applying the value of y in the second equation, we get

2x + 3[(9x + 12)/10]  =  13

(20x + 27x + 36)/10  =  13

47x + 36  =  130

47x  =  130 - 36

47x  =  94

x  =  94/47

x  =  2

Step 3 :

Now, we have to apply the value of x in the equation

y  =  (9 x + 12)/10

y  =  (9(2) + 12)/10

y  =  (18 + 12)/10

y  =  30/10

 y  =  3

So, the solution is (2, 3).

Example 4 :

A farmer plants corn and wheat on a 180-acre farm. The farmer wants to plant three times as many acres of corn as wheat. Write a system of linear equations that represents this situation. How many acres of each crop should the farmer plant?

Solution :

Let x be the number of acres that wheat has been planted.

Let y be the number of acres that corn has been planted.

x + y = 180

y = 3x

Applying the value of y, we get

x + 3x = 180

4x = 180

x = 180/4

x = 45

y = 3(45)

= 135

So, the number of acres of wheat is 45 and number of acres of corn is 135.

Example 5 :

A company that offers tubing trips down a river rents tubes for a person to use and “cooler” tubes to carry food and water. A group spends $270 to rent a total of 15 tubes. Write a system of linear equations that represents this situation. How many of each type of tube does the group rent?

Solution :

Let x be the number of 1 person tube, let y be the number of cooler tube.

Total number of tubes = 15

x + y = 15 ------(1)

20x + 12.50y = 270  ------(2)

y = 15 - x

Applying the value of y, we get

20x + 12.50(15 - x) = 270

20x + 187.5 - 12.50x = 270

7.5x + 187.5 = 270

7.5x = 270 - 187.5

7.5x = 82.5

x = 82.5/7.5

x = 11

Applying the value of x, we get

y = 15 - 11

y = 4

Number of one person tubes is 11 and number of cooler tubes is 4.

Example 6 :

A math test is worth 100 points and has 38 problems. Each problem is worth either 5 points or 2 points. How many problems of each point value are on the test?

Solution :

Total points = 100

Total number of problems = 38

Let x be the number of questions which gets 5 points.

Let y be the number of questions which gets 2 points.

5x + 2y = 100 ---------(1)

x + y = 38

y = 38 - x -----(2)

Applying the value of y in (1), we get

5x + 2(38 - x) = 100

5x + 76 - 2x = 100

3x = 100 - 76

3x = 24

x = 24/3

x = 8

Applying x = 8, we get

y = 38 - 8

y = 30

So, number of 5 point questions is 8, so number of 2 point questions is 30.

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