**Solving System of Linear Equation in Two-Variables**

Definition

Systems of Linear Equations

Systems of linear equations in two variables have either no solutions, one solution, or infinitely many solutions.

A system of equations with at least one solution is

consistent

.

A system with no solutions is

inconsistent

.

Graphing

Elimination

Substitution

QUIZ

ON YOUR OWN...

Exercise 2

Directions

: Determine the resulting equation by substituting the given value of one variable to each of the following equations. Then solve for the other variable using the resulting equation.

Equation Value of Variable

1. 4x + y = 7; y = x + 3

2. x + 3y = 12; x = 4 – y

3. 2x – 3y = 9; y = x – 2

4. 5x + 2y = 8; x = 3y + 1

5. 4x – 7y = -10; y = x – 4

6. -5x = y – 4; y = 3x + 5

Exercise 1

A set of linear equations in two variables is called a

system of linear equations

.

A

solution

of such a system is an ordered pair which is a solution of each equation in the system.

Example:

The ordered pair (4, 1) is a solution of the system since 3(4) + 2(1) = 14 and 2(4) – 5(1) = 3.

Example:

The ordered pair (0, 7) is not a solution of the system since 3(0) + 2(7) = 14 but 2(0) – 5(7) = – 35, not 3.

5. Which of the following ordered pairs satisfy both 2x + 7y > 5 and 3x – y ≤ 2?

a. (0, 0) b. (10, -1) c. (-4, 6) d. (-2, -8)

6. Which of the following is a graph of a system of linear inequalities in two variables?

a. c.

b. d.

7. Which of the following is a system of linear equations in two variables?

a. 2x – 7y = 8 c. x + 9y = 2

2x – 3y > 12

b. 3x + 5y = -2 d. 4x + 1 = 8

x – 4y = 9

DIRECTION: Choose the letter that you think best answers the question.

1. What point is the intersection of the graphs of the lines x + y = 8 and 2x – y = 1?

a. (1, 8) b. (3, 5) c. (5, 3) d. (2, 6)

2. If 2x + y = 9 and 2x – y = 11 , what is the value of x?

a. 4 b. 5 c. 10 d. 20

3. How many solutions does a consistent and independent system of

linear equations have?

a. 0 b. 1 c. 2 d. Infinite

4. Which system of equations has a graph that shows intersecting lines?

a. 2x + 4y = 14 b. -3x + y = 5

x + 2y = 7 6x – 2y = 1

c. 4x + 8y = 7 d. 3x + y = 10

x + 2y = 3 3x – y = 5

8. A restaurant owner would like to make a model which he can use as guide in writing a system of equations. He will use the system of equations in determining the number of kilograms of pork and beef that he needs to purchase daily given a certain amount of money (C), the cost (A) of a kilo of pork, the cost (B) of a kilo of beef, and the total weight of meat (D). Which of the following models should he make and follow?

a. Ax – By = C c. Ax + By = C

x + y = D x + y = D

b. Ax + By = C d. Ax – By = C

x – y = D x – y = D

9. Determine whether the given ordered pair is a solution of x + 3y = 8.

a. (2, 2) b. (-1, 3) c. (-2, -2) d. (1, 3)

10. Determine whether the given ordered pair is a solution of 15x + 5y = 75.

a. (0, 15) b. (-5, 15) c. (15, 0) d. (15, -5)

Three Ways to

Solve Systems of Linear

Equations in Two Variables

Directions

: Draw the graph of each equation in the system in one coordinate plane.

1. y = x + 7 2. y = 3x – 2

y = -2x + 1 8x + 7y = 15

3. 3x + 8y = 12 4. x – y = 6

8x – 5y = 12 2x + 7y = -6

**THANK YOU!**

I. Determine the resulting equation if one variable is solved

in terms of the other variable in one equation, and substitute this variable in the other equation. Then solve the system.

1. 3 x + y = 2 2. x = -y + 7

9x + 2y = 7 x – y = -9 5. 2x + 5y =9

-x + y = 2

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

4x + 3y = 20 3x – 2y = -5

II. Solve each system of linear equations by the elimination

method, then check your answers.

1. 3x + 2y = -4 2. 7x – 2y = 4

5x + y = 15 2x – y = -12 5. 5x + 2y = 6

-2x + y = -6

3. 2x + 3y = 7 4. x – 4y = 9

3x – 2y = 7 3x – 5y = 1