LINEAR AND QUADRATIC EQUATION SYSTEMS
LINEAR AND QUADRATIC EQUATION SYSTEMS
SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES
The general form of linear equations in two variables is :a1x + b1y = c1 , a1 ,b1 not both 0
a2x + b2y = c2 , a2 ,b2 not both 0
Solution set of linear equations in two variables can be found by :
1. Graphs
2. Elimination
3. Substitution
4. Combination of elimination and substitution
1. Solution by Graphs
To solve a system of linear equations in variable x and y by graphs, we must draw the two equations on the same coordinate system. Then find the point of intersections them. This point is called the solution of linear equations in two variables.
Examples :
1. By graph, find the solution set of 2x + y = 4
x – y = -1
Solution :
( Please, do it in millimeter paper)
Step 1. Draw the two equations on the same coordinate system
If , then system of linear equations in two variables have only one solution and the graphs intersect in one and only one point.
2. By the graph, find the solution set of x + 2y = 4
Step 2 . The system have no solution, because the graphs have no intersection point
Conclude :
If , then system of linear equations in two variables have no solution
And the graphs are distinct parallel lines.
3. By the graph, find the solution set of x – y = 2
2x – 2y = 4
Solution :
Step 1. x – y = 2 2x – 2y = 4
x x
y y
The graph:
Step 2. The system have infinitely many solutions, because the graphs have many
Intersection point
Conclude :
If , then system of linear equations in two variables have infinitely
Many solutions and the graphs are the same line.
2. Solution by Elimination
There are two steps :
a. eliminate one variable
b. eliminate other variable
Examples :
1. By elimination, find the solution set of 2x – y = 4
3x + 2y = 13
Solution :
2x – y = 4 …………..(1)
3x + 2y = 13………...(2)
Elimination of y:
4x – 2y = 8………….(3) ( multiply (1) by 2)
3x + 2y = 13
+
7x = 21 (addition of (3) and (2) so that eliminate the variable y)
x = 3
Elimination of x :
6x – 3y = 12…………(4) (multiply (1) by 3)
6x + 4y = 26…………(5) (multiply (2) by 2)
-7y = -14 (subtraction (4) and (5) so that eliminate the
y = 2 variable x)
# The solution set is
2. By elimination, find the solution set of 2x + y = 4
x – y = -1
Solution :
2. Solution by Substitution
There are two steps :
a. Take one equation and express one variable with other variable
b. Substitute to other equation
Examples :
1. By substitution, find the solution set of x + y = 4
4x + 3y = 13
Solution :
x + y = 4……..(1) y = 4 – x …….(3) ( express y with x)
4x + 3y = 13 ……(2)
Substitute (3) to (2)
4x + 3y = 13
4x + 3(4- x)= 13
4x + 12 – 3x = 13
x = 1
Substitute x = 1 to (3)
y = 4 – x
= 4 – 1
= 3
# The solution set is
2. By substitution, find the solution set is 2x + y = 8
3x + 2y = 13
Solution :
4. Solution by Combination of Elimination and Substitution
Examples :
1) By combination of elimination and substitution, find the solution set of
4x + 3y = 10
2x + y = 4
Solution : 4x + 3y = 10 x 1 4x + 3y = 10
2x + y = 4 x 3 6x + 3y = 12
-
- 2x = - 2
x = 1
Substitute x = 1 to equation which easy, example : 2x + y = 4
2.1 + y = 4
y = 2
☻ The solution set is
2) By combination of elimination and substitution, find the solution set of
Solution :
EXERCISES :
1. Find the solution set of the following linear equation systems by graphs
a. 3x –y = 2 b. 2x + 8y = 6 c. 3x – y = 3
x + y = 2 4x + 6y = 12 6x – 2y = 12
2. Find the solution set of the following linear equation systems by elimination
a. 2x + 11y = -1 b. 5x = 2y + 4 c. –x + 2y + 1 = 0
3x – 7y = 22 x – 4y = -10 2x + 3y + 3 = 0
3. Find the solution set of the following linear equation systems by substitution
a. b. c.
4. Find the solution set of the following linear equation systems by combination of elimina-
tion and substitution
a. d. g.
b. e. h. c. f. i
5. Find point intersection of the lines y = 2x – 2 and x – y +1 = 0 and sketch it
6. Given three resistors, they are resistor A = R1 ohms, resistor B = R2 ohms and resistor
C = R2 ohms. When resistors A and B are serried, the shunt resistor is 10 ohms. When
Resistors A, B, and C are serried the shunt resistor is 13 ohms. Find shunt resistor when
Resistors A, B, and C are parallelized
7. Mr. Agus works for 6 days which 4 days are overtime to get 74,000 rupiah. Mr. Bardi
works for 5 days which 2 days are overtime to get 55,000 rupiah. Mr. Agus, Mr. Bardi,
and Mr. Dodo work under the same payment system. If Mr. Dodo works for 5 days over-
time, then the payment that he shall receive is ….
8. In the year 2002 the age of a girl is equal to a quarter of her mother’s age. In the year
2006 the age of the girl is one third her mother’s age. Find the year when the girl born
9. If the numerator of fraction is added 2 and the denominator is added 5 then the value of
fraction is a half. If the numerator minus one and the denominator is added one then the
value of fraction is a third. Find the fraction.
10. The price of eggs and meat in two shops are shown in the following table :
Eggs (kg)
Meat (kg)
Price ( Thousand rupiahs )
Shop A
100
200
6900
Shop B
60
70
2640
Find the price of one kg egg and one kg meat in two shops
B. SYSTEM OF LINEAR EQUATIONS IN THREE VARIABLES
The general form is :
a1x + b1y + c1z = d1 ; a1, b1, c1 not both 0
a2x + b2y + c2z = d2 ; a2, b2, c2 not both 0
a3x + b3y + c3z = d3 ; a3, b3, c3 not both 0
Solution set of linear equations in three variables can be found by :
a. Substitution
b. Combination of elimination and substitution
1. Solution by substitution
Example :
Find the solution set of
Solution :
Equation (1) is made become y = 7 – 2x – 3z …………(4)
Substitute equation (4) to equation (2) so :
x + 2 ( 7 – 2x – 3z ) + z = 1
x + 14 – 4x – 6z + z = 1
- 3x – 5z = - 13
3x + 5z = 13 …………..(5)
Substitute equation (4) to equation (3) so :
……………………………………………..
……………………………………………..
……………………………………………..
z = 3 – x ……….(6)
Substitute equation (6) to equation (5) so :
………………………………………………
………………………………………………
………………………………………………
x = ….
Substitute x =… to equation (6) so : z = ….
Substitute x = … and z = …. To equation (4) so y = ….
☻Solution set is
2. Solution set by combination of substitution and elimination
Example :
Find solution set of
Solution :
Elimination z of equation (1) and (2)
2x + y –z = 5
x + 2y +2z = 13
………………..
…………………………………………(4)
Elimination z of equation (1) and (3)
2x +y –z = 5 x 2 …………………………
x – 3y + 2z = -4 x1 …………………………
+
…………………………...............(5)
Elimination y of equation (4) and (5)
…………………..
…………………..
-
……………………
x = …..
Substitute x = …. To equation (4)
……………………..
……………………..
y = ….
Substitute x = … and y = … to equation (1)
……………………..
……………………..
z = ….
☻The solution set is
EXERCISES
1. Find the solution set of the following linear equation systems by substitute
a. b. c.
2. Find the solution set of the following linear equation systems by combination of substitution
and elimination
a. b. c.
d. e.
3. The price of two bananas, two guavas, and one mango is 1,400 rupiahs. The price of one
banana, one guava, and two mangoes is 1,300 rupiahs. The price of one banana, three
guavas, and one mango is 1,500 rupiahs. Find the price of one banana, one guava and one
mango.
4. The graph of quadratic function y = ax2 + bx + c passes through the points (-1,0),(1,6) and
(2,12). Find a, b, c, then give the equation of the graph.
5. The circle x2 + y2 + Ax + By +C = 0, passes through the points (3,-1),(5,3) and (6,2). Find
A, B, C and then give the equation.
6. The quadratic form ax2 + bx + c has value -1 if x = 1, its value 4 if x = 2, and has value 17
if x = 3. Find value a, b, and c
C. SIMULTANEOUS EQUATIONS, ONE LINEAR – ONE QUADRATIC.
AND SIMULTANEOUS EQUATION, TWO QUADRATICS
1. Solution set by substitute
Example :
a) Find the solution set of
Solution :
Substitute equation (1) to (2)
x2 +1 = x + 3
…………………….
…………………….
x = …. or x = ….
For x = …., then y = ….. and the point is (….,….)
For x = …., then y = ….. and the point is (….,….)
☻Solution set is
NOTE :
The solution set is the point intercept of (1) and (2).
By the graph can be expressed as follows :
Y
y = x2 + 1
y = x + 3
(2,5)
(-1,2)
X
b) Find the solution set of
Solution :
2x2 – 3x + 1 = x2 + x + 6
x2 – 4x – 5 = 0
(x + 1)(x – 5) = 0
x = -1 or x = 5
For x = -1 then y = 6
For x = 5 then y = 36
☻Solution set is
2. Solution set by factoring of the quadratic form
Example :
Find the solution set of
Solution :
Factorize the equation (2)
x2 + 2xy + y2 -25 = 0
(x + y + 5)(x + y – 5) = 0
x + y + 5 = 0 or x + y – 5 = 0
For x + y + 5 = 0 then y = - x – 5
Substitute y = - x – 5 to (1) :
(- x – 5) – x = 3
-2x = 8
x = -4 then y = -1, so the point is (-4,-1)
For x + y – 5 = 0 then y = 5 – x
Substitute y = 5 – x to (1) :
(5 – x) – x = 3
-2x = -2
x = 1 then y = 4, so the point is (1,4)
☻Solution set is
EXERCISES
1. Find the solution set of the following equation system :
a. d.
b. e.
c. f.
2. Find the border value of m so that the line y = x – 10 and curve y = x2 – mx + 6 have two
distinct intercept point
3. Find value of m so that the line y = mx – 9 touches the quadratic function y = x2
4. Given the equation system
Find the border value of m so that the equation system has two point in solution
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