SYSTEMS OF LINEAR EQUATIONS

AND INEQUALITIES

Week 04: Solving System of Linear Equations Algebraically Prepared by: Jojo M. Lucion

DISCUSSION

To solve a system of linear equations algebraically, we must reduce the system to a single equation with only one variable.

Do you still remember the properties of equality?

Let us call the properties as the Golden Rule of Equations.

Whatever you do unto one side of an equation, do the same thing unto the other side of

the equation. Then the equation will remain true. In particular, you can:

• Add the same number to both sides - A.P.E.

• Subtract the same number from both sides - S. P. E.

• Multiply both sides by the same number - M.P.E.

• Divide both sides by the same number -D.P.E.

Let us recall how to transform an equation with two variables to y = mx + b

Transform each of the following equations to y = mx + b

Now , are you ready to solve system of linear equations using the algebraic method?

A. Using the Substitution Method

Example 1

Solve the following system of equations:

x + 2y = 7 E. 1 y - 1

= 2x E. 2

Solution:

• Solve E. 2 for y:

y = 2x + 1

• Eliminate y in E.1 by substituting 2x + 1 for y:

x + 2y = 7

x + 2(2x + 1)= 7

x + 4x + 2 = 7

5x + 2 = 7

5x = 7-2

5x = 5

x = 5 =1 5

• Find the value of y when x = 1 by substituting 1 for x in either of the two original

equations: x + 2y = 7

1 + 2y = 7

2y = 7 - 1

2y = 6

y= 6 y= 3 Solution(1, 3) 2

B Using the Addition Method

Example 1

Solve by the addition method

3x + y = 10 E. 1

2x - y = 5 E. 2

C.Using Subtraction Method

D.Using Multiplication with the Addition Method

Solution :

Example 2

Solve: 2x + 3y = -1 E. 1

5x-2y = -12 E.2