Exercise 20 Page &nbsp; 402

(II):Subtract terms

x=5-y -12y=30

DivEqn

(II):.LHS /(-12).=.RHS /(-12).

x=5-y y=-2.5

Now that we have isolated the y-variable, let's substitute it into the first equation to solve for the x-variable.

x=5-y & (I) y=-2.5 & (II)

(I):y= -2.5

x=5-( -2.5) y=-2.5

(I):Subtract terms

x=7.5 y=-2.5

The system has one solution, (7.5,-2.5).

b The first equation has variables with a coefficient of 1.Let's start by isolating the x-variable in the first equation before substituting it into the second equation.

x+y=7 & (I) 5x+2y=23 & (II)

(I):LHS-y=RHS-y

x=7-y 5x+2y=23

Now that we have isolated the x-variable, let's substitute it into the second equation to solve for the y-variable.

x=7-y & (I) 5x+2y=23 & (II)

(II):x= 7-y

x=7-y 5( 7-y)+2y=23

Distr

(II):Distribute 5

x=7-y 35-5y+2y=23

AddTerms

(II):Add terms

x=7-y 35-3y=23

(II):LHS-35=RHS-35

x=7-y -3y=-12

DivEqn

(II):.LHS /(-3).=.RHS /(-3).

x=7-y y=4

Now that the y-variable is solved, let's substitute the value into the first equation and solve for the x-variable.

x=7-y y=4

(I):y= 4

x=7- 4 y=4

(I):Subtract terms

x=3 y=12

The system has one solution, (3,12).

c Notice that in the first equation, the y-variable has a coefficient of 1. We can start by isolating the y-variable in the first equation before substituting it into the second equation.

3x+y=5 & (I) 6x+2y=12 & (II)

(I):LHS-3x=RHS-3x

y=5-3x 6x+2y=12

Now that we have isolated the y-variable, let's substitute it into the second equation to solve for the x-variable.

y=5-3x & (I) 6x+2y=12 & (II)

(II):y= 5-3x

y=5-3x 6x+2( 5-3x)=12

Distr

(II):Distribute 2

y=5-3x 6x+10-6x=12

(II):Subtract terms

x=7-y 10≠ 12

Since 10 is not equal to 12 it is a false statement and there are no solutions.

d In the second equation, the x-variable has a coefficient of 1. This leads us to start by isolating the x-variable in the second equation before substituting it into the first equation.

2x+5y=-12 & (I) x+7y=-15 & (II)

(II):LHS-7y=RHS-7y

2x+5y=-12 x=-15-7y

Now that we have isolated the x-variable, let's substitute it into the first equation to solve for the y-variable.

2x+5y=-12 & (I) x=-15-7y & (II)