Systems of Linear Equations I

System of Equations

Related topics:1, 9, 10.

A system of two linear equations in two variables is called the system formed by two equations:

Ax + By = C
Ex + Fy = G

There are two main techniques for solving this kind of systems.

I. The Substitution Method

Consider the following system:

x + y = 2
2x - y = 7

1. Solve one of the equations for one variable in terms of the other, try to avoid fractions.

x + y = 2
y = -x + 2

2. Substitute the expression above to the other equation.

2x - y = 7
2x - (-x + 2) = 7
2x + x - 2 = 7
3x - 2 = 7
3x = 9
x = 3

3. Substitute the solution above to one of the equations in the system
and solve.

x + y = 2
3 + y = 2
y = 2-3
y = -1

4. Check to see if the solutions obtained are right.

x + y = 2
3 - 1 = 2
2x - y = 7
2(3) - (-1) = 7
6 + 1 = 7

x + y = 2
2x - y = 7

1. Multiply one of the equation with an appropriate real number that will eliminate one of the variables when the equations are added together.
Multiply the first equation with 1 and add the first equation to the second.

x + y = 2
2x - y = 7
3x      =9

Solve 3x=9 ; get x=3.

2. Substitute x = 3 to one of the equation to find the value of y.

x = 3
x + y = 2
3 + y = 2
y = -1

3. Check the solutions with the equations.

x + y = 2
3 - 1 = 2
2x - y = 7
2(3) - (-1) = 7
6 + 1 = 7

System of Three Equations in Three Variables:

Ax + By +Cz = D
Ex + Fy +Gz = H
Ix  + Jy  + Kz= L

Solving system of three equations in three variables is similar to solving system with two variables. The solution set is a set of ordered triple of real numbers (x,y,z).

Example:

2x + 3y + z = 1
5x + 2y - 3z= 8
x  -  4y -  z = 18

Multiply the first equation by 3 and add the result to the second equations:

3(2x + 3y + z) = 3
5x + 2y - 3z = 8
11x + 11y      = 11    factor 11 out and get

x + y =1

Now, add the first and third equations.

2x + 3y + z = 1
x  - 4y -  z = 18
3x -  y       =19

x + y = 1
3x - y = 19
4x     =20
x = 5

Substitute x = 5 into one of the two new equations, to obtain y:

x + y = 1
5 + y = 1
y = -4

Substitute y = -4 and x = 5 into one of the original three equations to get z:

x - 4y - z = 18
5 - 4(-4) - z = 18
5 + 16 - z = 18
5 + 16 - 18 = z
3 = z

The solution set is { 5,-4,3}.

EXERCISES

Solve the following systems by substitution:

1.  x  -   2y = 0
3x + 2y = 8

2.  x + 6y = 19
x - 7y = -7

3.   8x + 5y = 100
9x - 10y = 50

4.   2x + 5y = 29
5x + 2y = 13

5. A combined total of \$12000 is invested in two bonds that pay 8.5% and 10% simple interest. The annual total interest is \$1140. How much is invested in each bond?

6. Find the 2 integers satisfying the following: the sum of the larger number and twice the smaller number is 61 and the difference of the two numbers is 7.

Solve the following systems by elimination:

7.    x  - 3y = 7
-2x + 6y = -14

8.    4x + 3y = 8
x - 2y = 13

9.   3x - 2y = -20
5x + 6y = 32

10.   -2x + 5y = 5
4x - 10y = 9

11.    x + y + z = -3
4x + y - 3z = 11
2x - 3y + 2z = 9

12.   x + y - z = 2
x + 2y - 4z = 3
-2x - 2y + 2z = 5

13.   x + 6y + 2z = 9
3x - 2y + 3z = -1
5x - 5y + 2z = 7