Chapter 6.3, Problem 46PFA

a.

To determine

To define the variables

Explanation of Solution

Given:

There are two fractions that is greater than the other.

The sum of the two fractions is $\frac{11}{12}$ .

The difference of the fractions is $\frac{5}{12}$ .

Let x be the greater fraction and y be the other variable.

b.

To determine

To write the system of equations

Answer to Problem 46PFA

  $12x+12y=11$

  $12x-12y=5$

Explanation of Solution

Given:

There are two fractions that is greater than the other.

The sum of the two fractions is $\frac{11}{12}$ .

The difference of the fractions is $\frac{5}{12}$ .

Calculation:

Given that the sum of the two fractions is $\frac{11}{12}$ ,

  $\Rightarrow x+y=\frac{11}{12}$

In a simplified form,

  $\Rightarrow 12x+12y=11$

Also, the difference of the fractions is $\frac{5}{12}$ .

  $\Rightarrow x-y=\frac{5}{12}$

In a simplified form,

  $\Rightarrow 12x-12y=5$

Conclusion:

The system of equations are $12x+12y=11$ and $12x-12y=5$

c.

To determine

To find the fractions

Answer to Problem 46PFA

Greater Fraction: $\frac{2}{3}$

Lesser Fraction: $\frac{1}{4}$

Explanation of Solution

Given:

From subpart (b),

  $12x+12y=11$

  $12x-12y=5$

Calculation:

Let,

  $12x+12y=11\to \text{(1)}$

  $12x-12y=5\to \text{(2)}$

Add equations (1) and (2).

  $\begin{array}{l}\Rightarrow (12x+12y)+(12x-12y)=11+5\\ \Rightarrow 12x+12x+12y-12y=16\\ \Rightarrow 24x=16\\ \Rightarrow x=\frac{16}{24}\\ \Rightarrow x=\frac{2}{3}\end{array}$

Substitute the value of x in equation (1).

  $\begin{array}{l}\Rightarrow 12(\frac{2}{3})+12y=11\\ \Rightarrow \frac{24}{3}+12y=11\\ \Rightarrow 8+12y=11\\ \Rightarrow 12y=11-8\\ \Rightarrow y=\frac{3}{12}\\ \Rightarrow y=\frac{1}{4}\end{array}$

Conclusion:

Therefore, the greater fraction is $x=\frac{2}{3}$ and lesser fraction is $y=\frac{1}{4}$ .

d.

To determine

To explain how to use elimination to solve the system of equations

Explanation of Solution

Assume two variables, one is greater function and another one is lesser fraction.

Simplify and modify the equations such that at least one of the variables get eliminated.

If one of the variables is eliminated, the other variable can be found.

Substituting the value of one variable in any one of the equations, the other variable also can be found.

Compare both the variables and find the greater of them.

The greater is the greater fraction and the other is the lesser one.