<< Chapter < Page Chapter >> Page >

Substitution and parallel lines

If computations eliminate all the variables and produce a contradiction, the two lines of a system are parallel, and the system is called inconsistent.

Sample set b

Solve the system { 2 x y = 1 4 x 2 y = 4 ( 1 ) ( 2 )

Step 1:  Solve equation 1 for y .
      2 x y = 1 y = 2 x + 1 y = 2 x 1

Step 2:  Substitute the expression 2 x 1 for y into equation 2.
      4 x 2 ( 2 x 1 ) = 4

Step 3:  Solve the equation obtained in step 2.
      4 x 2 ( 2 x 1 ) = 4 4 x 4 x + 2 = 4 2 4

Computations have eliminated all the variables and produce a contradiction. These lines are parallel.
A graph of two parallel lines. One line is labeled with the equation two x minus y is equal to one and passes through the points one, one, and zero, negative one. A second line is labeled with the equation four x minus two y is equal to four and passes through the points one, zero, and zero, negative two.
This system is inconsistent.

Got questions? Get instant answers now!

Practice set b

Slove the system { 7 x 3 y = 2 14 x 6 y = 1

Substitution produces 4 1 , or 1 2 2 , a contradiction. These lines are parallel and the system is inconsistent.

Got questions? Get instant answers now!

Substitution and coincident lines

The following rule alerts us to the fact that the two lines of a system are coincident.

Substitution and coincident lines

If computations eliminate all the variables and produce an identity, the two lines of a system are coincident and the system is called dependent.

Sample set c

Solve the system { 4 x + 8 y = 8 3 x + 6 y = 6 ( 1 ) ( 2 )

Step 1:  Divide equation 1 by 4 and solve for x .
      4 x + 8 y = 8 x + 2 y = 2 x = 2 y + 2

Step 2:  Substitute the expression 2 y + 2 for x in equation 2.
      3 ( 2 y + 2 ) + 6 y = 6

Step 3:  Solve the equation obtained in step 2.
      3 ( 2 y + 2 ) + 6 y = 6 6 y + 6 + 6 y = 6 6 = 6

Computations have eliminated all the variables and produced an identity. These lines are coincident.
A graph of two coincident lines. The line is labeled with the equation x plus two y is equal to two and a second label with the equation three x plus six y is equal to six. The lines pass through the points zero, one and two, zero. Since the lines are coincident, they have the same graph.
This system is dependent.

Got questions? Get instant answers now!

Practice set c

Solve the system { 4 x + 3 y = 1 8 x 6 y = 2

Computations produce 2 = 2 , an identity. These lines are coincident and the system is dependent.

Got questions? Get instant answers now!

Systems in which a coefficient of one of the variables is not 1 or cannot be made to be 1 without introducing fractions are not well suited for the substitution method. The problem in Sample Set D illustrates this “messy” situation.

Sample set d

Solve the system { 3 x + 2 y = 1 4 x 3 y = 3 ( 1 ) ( 2 )

Step 1:  We will solve equation ( 1 ) for y .
      3 x + 2 y = 1 2 y = 3 x + 1 y = 3 2 x + 1 2

Step 2:  Substitute the expression 3 2 x + 1 2 for y in equation ( 2 ) .
      4 x 3 ( 3 2 x + 1 2 ) = 3

Step 3:  Solve the equation obtained in step 2.
      4 x 3 ( 3 2 x + 1 2 ) = 3 Multiply both sides by the LCD ,  2 . 4 x + 9 2 x 3 2 = 3 8 x + 9 x 3 = 6 17 x 3 = 6 17 x = 9 x = 9 17

Step 4:  Substitute x = 9 17 into the equation obtained in step 1 , y = 3 2 x + 1 2 .
      y = 3 2 ( 9 17 ) + 1 2
      y = 27 34 + 17 34 = 10 34 = 5 17
     We now have x = 9 17 and y = 5 17 .

Step 5:  Substitution will show that these values of x and y check.

Step 6:  The solution is ( 9 17 , 5 17 ) .

Got questions? Get instant answers now!

Practice set d

Solve the system { 9 x 5 y = 4 2 x + 7 y = 9

These lines intersect at the point ( 1 , 1 ) .

Got questions? Get instant answers now!

Exercises

For the following problems, solve the systems by substitution.

{ 3 x + 2 y = 9 y = 3 x + 6

( 1 , 3 )

Got questions? Get instant answers now!

{ 5 x 3 y = 6 y = 4 x + 12

Got questions? Get instant answers now!

{ 2 x + 2 y = 0 x = 3 y 4

( 1 , 1 )

Got questions? Get instant answers now!

{ 3 x + 5 y = 9 x = 4 y 14

Got questions? Get instant answers now!

{ 3 x + y = 4 2 x + 3 y = 10

( 2 , 2 )

Got questions? Get instant answers now!

{ 4 x + y = 7 2 x + 5 y = 9

Got questions? Get instant answers now!

{ 6 x 6 = 18 x + 3 y = 3

( 4 , 1 3 )

Got questions? Get instant answers now!

{ x y = 5 2 x + y = 5

Got questions? Get instant answers now!

{ 5 x + y = 4 10 x 2 y = 8

Dependent (same line)

Got questions? Get instant answers now!

{ x + 4 y = 1 3 x 12 y = 1

Got questions? Get instant answers now!

{ 4 x 2 y = 8 6 x + 3 y = 0

( 1 , 2 )

Got questions? Get instant answers now!

{ 2 x + 3 y = 12 2 x + 4 y = 18

Got questions? Get instant answers now!

{ 3 x 9 y = 6 6 x 18 y = 5

inconsistent (parallel lines)

Got questions? Get instant answers now!

{ x + 4 y = 8 3 x 12 y = 10

Got questions? Get instant answers now!

{ x + y = 6 x y = 4

( 1 , 5 )

Got questions? Get instant answers now!

{ 2 x + y = 0 x 3 y = 0

Got questions? Get instant answers now!

{ 4 x 2 y = 7 y = 4

( 15 4 , 4 )

Got questions? Get instant answers now!

{ x + 6 y = 11 x = 1

Got questions? Get instant answers now!

{ 2 x 4 y = 10 3 x = 5 y + 12

( 1 , 3 )

Got questions? Get instant answers now!

{ y + 7 x + 4 = 0 x = 7 y + 28

Got questions? Get instant answers now!

{ x + 4 y = 0 x + 2 3 y = 10 3

( 4 , 1 )

Got questions? Get instant answers now!

{ x = 24 5 y x 5 4 y = 3 2

Got questions? Get instant answers now!

{ x = 11 6 y 3 x + 18 y = 33

inconsistent (parallel lines)

Got questions? Get instant answers now!

{ 2 x + 1 3 y = 4 3 x + 6 y = 39

Got questions? Get instant answers now!

{ 4 5 x + 1 2 y = 3 10 1 3 x + 1 2 y = 1 6

( 1 , 1 )

Got questions? Get instant answers now!

{ x 1 3 y = 8 3 3 x + y = 1

Got questions? Get instant answers now!

Exercises for review

( [link] ) Find the quotient: x 2 x 12 x 2 2 x 15 ÷ x 2 3 x 10 x 2 2 x 8 .

( x 4 ) 2 ( x 5 ) 2

Got questions? Get instant answers now!

( [link] ) Find the difference: x + 2 x 2 + 5 x + 6 x + 1 x 2 + 4 x + 3 .

Got questions? Get instant answers now!

( [link] ) Simplify 81 x 8 y 5 z 4 .

9 x 4 y 2 z 2 y

Got questions? Get instant answers now!

( [link] ) Use the quadratic formula to solve 2 x 2 + 2 x 3 = 0.

Got questions? Get instant answers now!

( [link] ) Solve by graphing { x y = 1 2 x + y = 5
An xy coordinate plane with gridlines labeled negative five and five with increments of one unit for both axes.

( 2 , 1 )
A graph of two lines intersecting at a point with coordinates negative two, one. One of the lines is passing through a point with coordinates zero, five and the other line is passing through a point with coordinates zero, negative one.

Got questions? Get instant answers now!

Questions & Answers

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3
Esther Reply
derivative of logarithms function
Iqra Reply
how to solve this question
sidra
ex 2.1 question no 11
khansa
anyone can help me
khansa
question please
Rasul
ex 2.1 question no. 11
khansa
i cant type here
khansa
Find the derivative of g(x)=−3.
Abdullah Reply
any genius online ? I need help!!
Guzorochi Reply
how can i help you?
Pina
need to learn polynomial
Zakariya
i will teach...
nandu
I'm waiting
Zakariya
plz help me in question
Abish
How can I help you?
Tlou
evaluate the following computation (x³-8/x-2)
Murtala Reply
teach me how to solve the first law of calculus.
Uncle Reply
teach me also how to solve the first law of calculus
Bilson
what is differentiation
Ibrahim Reply
only god knows😂
abdulkadir
f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask