Chapter ISG, Problem 6.2.1P

a.

To determine

To solve system of equations.

Answer to Problem 6.2.1P

  $x=12$

  $y=18$

Explanation of Solution

Given information:

  $\begin{array}{l}x=y-6\\ 2x-y=6\end{array}$

Calculation:

Let us assume the equations be (1) and (2) respectively,

Taking (1)

  $x=y-6$

Put value of x in (2)

  $2(y-6)-y=6$

Solving,

  $\begin{array}{l}2y-12-y=6\\ y=6+12\end{array}$

So,

  $y=18$

Put value of y in (1)

  $x=18-6$

So,

  $x=12$

Hence,

  $x=12$ and $y=18$

b.

To determine

To solve the system of equation

Answer to Problem 6.2.1P

  $x=1$ and $y=4$

Explanation of Solution

Given information:

  $\begin{array}{l}x-3y=-13\\ x+y=3\end{array}$

Calculation:

Let equations be (1) and (2) respectively,

Taking (2)

  $x=3-y$  (3)

Put value of x in (1)

  $3-y-3y=-13$

Solving,

  $-4y=-16$

Dividing by -4,

  $y=4$

Put value of y in (3)

  $x=3-4$

So,

  $x=1$

Hence, $x=1$ and $y=4$

c.

To determine

To solve the system of equation

Answer to Problem 6.2.1P

  $x=1.3$ andy = $-0.2$

Explanation of Solution

Given information:

  $\begin{array}{l}2x+y=2.4\\ x-2y=1.7\end{array}$

Calculation:

Let us assume equation be (1) and (2)

Taking (2)

  $x-2y=1.7$

Rearranging,

  $x=1.7+2y$   -(3)

Putting value of $x$ in (1)

  $2(1.7+2y)+y=2.4$

Solving,

  $\begin{array}{l}3.4+4y+y=2.4\\ 5y=2.4-3.4\end{array}$

So,

  $5y=-1$

Dividing by 5,

  $y=\frac{-1}{5}$

y = $-0.2$

Put in (3)

  $x=1.7+2(-0.2)$

Solving,

  $x=1.7-0.4$

So,

  $x=1.3$

Hence, $x=1.3$ and y = $-0.2$

d.

To determine

To solve the system of equation

Answer to Problem 6.2.1P

No solution.

Explanation of Solution

Given information:

  $\begin{array}{l}2x-3y=26\\ 3x-4.5y=39\end{array}$

Calculation:

Let us assume equation be (1) and (2)

Taking (1)

  $2x-3y=26$

Rearranging,

  $2x=26+3y$

So,

  $x=\frac{26+3y}{2}$  -(3)

Putting value of $x$ in (2)

  $3(\frac{26+3y}{2})-4.5y=39$

Solving,

  $\begin{array}{l}\frac{102+9y}{2}-4.5y=39\\ 51+4.5y-4.5y=39\end{array}$

Here, y terms are cancelled out, it system has no solution.

e.

To determine

To solve the system of equations

Answer to Problem 6.2.1P

  $x=2$ and $y=4$

Explanation of Solution

Given information:

  $\begin{array}{l}2x+3y=16\\ 3x+6y=30\end{array}$

Calculation:

Assume equations to be (1) and (2)

Taking (1),

  $2x+3y=16$

Solving,

  $2x=16-3y$

So,

  $x=\frac{16-3y}{2}$  -(3)

Put x in (2)

  $3(\frac{16-3y}{2})+6y=30$

Taking LCM and multiplying,

  $48-9y+12y=30\times 2$

So,

  $3y=60-48$

And,

  $3y=12$

Dividing equation by 3,

  $y=4$

Put value of y in (3)

  $x=\frac{16-3(4)}{2}$

Solving,

  $x=\frac{16-12}{2}$

So,

  $x=2$

Hence, $x=2$ and $y=4$

f.

To determine

To solve the system of equations

Answer to Problem 6.2.1P

  $x=0.5$ and $y=9$

Explanation of Solution

Given information:

  $\begin{array}{l}4x+3y=25\\ 3x=x+2y-19\end{array}$

Calculation:

Making it into standard equation,

  $\begin{array}{l}4x+3y=25\\ 2x-2y=-19\end{array}$

Assuming equations as (1) and (2),

Taking (1)

  $4x+3y=25$

Rearranging,

  $4x=25-3y$

So,

  $x=\frac{25-3y}{4}$  -(3)

Put value of x in (2)

  $2(\frac{25-3y}{4})-2y=-19$

Solving,

  $25-3y-4y=-19\times 2$

So,

  $25-3y-4y=-38$

And,

  $-7y=-63$

Dividing equation by -7,

  $y=9$

Put y in (3)

  $x=\frac{25-3(9)}{4}$

Solving,

  $x=\frac{25-27}{4}$

Dividing equation,

  $x=0.5$

Hence, $x=0.5$ and $y=9$ is the solution

g.

To determine

To solve the system of equation

Answer to Problem 6.2.1P

y= 2 and x =1

Explanation of Solution

Given information:

  $\begin{array}{l}2y+3x=7\\ 3x-2y=-1\end{array}$

Calculation:

Let the equations be (1) and (2) respectively,

Taking (1)

  $2y+3x=7$

Rearranging,

  $2y=7-3x$

So,

  $y=\frac{7-3x}{2}$   -(3)

Put y in (2)

  $3x-2(\frac{7-3x}{2})=-1$

Solving,

  $3x-7+3x=-1$

So,

  $6x=-1+7$

Dividing equation by 6,

  $x=1$

Put value of x in (3)

y = $\frac{7-3(1)}{2}$

Solving,

  $y=\frac{4}{2}$

Hence, y= 2 and x =1 are the solution.

h.

To determine

To solve the system of equations

Answer to Problem 6.2.1P

  $y=0$ and $x=6$

Explanation of Solution

Given information:

  $\begin{array}{l}\frac{4}{3}x+y=8\\ x=3y+6\end{array}$

Calculation:

Let the equations be (1) and (2) respectively,

Taking (1),

  $\frac{4}{3}x+y=8$

Rearranging,

  $y=8-\frac{4}{3}x$   -(3)

Put y in (2)

  $x=3(8-\frac{4}{3}x)+6$

Solving,

  $x=24-4x+6$

So,

  $5x=30$

Dividing equation by 5,

  $x=6$

Put value of x in (3)

  $y=8-\frac{4}{3}(6)$

Solving,

  $y=8-8$

Hence, $y=0$ and $x=6$ is the solution of the equation