Chapter 3, Problem 6CA

a.

To determine

To solve the equation $2x-9=5x-33$ and classify the equation based on the solution.

Answer to Problem 6CA

  $x=8$ , conditional equation

Explanation of Solution

Given:

  $2x-9=5x-33$

Concept used:

1. If an equation is true for all values of the variable, then it is an identity equation.
2. If an equation is true for only some values of the variable, then it is a conditional equation.
3. If an equation results in a false statement, then it is an inconsistent equation.

Calculation:

  $2x-9=5x-33$

Solving the equation,

  $\begin{array}{l}\Rightarrow 2x-5x=-33+9\\ \Rightarrow -3x=-24\\ \Rightarrow 3x=24\\ \Rightarrow x=8\end{array}$

Now, classifying the equation on the basis of the solution obtained.

Since the solution set consists of one number: {8}, it is the only solution and, therefore, the given equation is conditional.

Conclusion:

Therefore, the solution to the given equation is $x=8$ and it is a conditional equation.

b.

To determine

To solve the equation $5x-6=10x+10$ and classify the equation based on the solution.

Answer to Problem 6CA

  $x=-\frac{16}{5}$, conditional equation

Explanation of Solution

Given:

  $5x-6=10x+10$

Concept used:

1. If an equation is true for all values of the variable, then it is an identity equation.
2. If an equation is true for only some values of the variable, then it is a conditional equation.
3. If an equation results in a false statement, then it is an inconsistent equation.

Calculation:

  $5x-6=10x+10$

Solving the equation

  $\begin{array}{l}\Rightarrow 5x-10x=10+6\\ \Rightarrow -5x=16\\ \Rightarrow x=-\frac{16}{5}\end{array}$

Now, classifying the equation on the basis of the solution obtained.

Since the solution set consists of one number: $\left\{-\frac{16}{5}\right\}$ , it is the only solution and, therefore, the given equation is conditional.

Conclusion:

Therefore, the solution to the given equation is $x=-\frac{16}{5}$ and it is a conditional equation.

c.

To determine

To solve the equation $2\left(8x-3\right)=4\left(4x+7\right)$ and classify the equation based on the solution.

Answer to Problem 6CA

Inconsistent equation

Explanation of Solution

Given:

  $2\left(8x-3\right)=4\left(4x+7\right)$

Concept used:

1. If an equation is true for all values of the variable, then it is an identity equation.
2. If an equation is true for only some values of the variable, then it is a conditional equation.
3. If an equation results in a false statement, then it is an inconsistent equation.

Calculation:

  $2\left(8x-3\right)=4\left(4x+7\right)$

Solving the equation

  $\begin{array}{l}\Rightarrow 16x-6=16x+28\\ \Rightarrow 16x-6-16x=16x+28-16x\\ \Rightarrow -6\ne 28\end{array}$

[Subtracting $16x$ from both sides]

Since it is a false statement, there is no solution to the given equation and hence, it is an inconsistent equation.

Conclusion:

Therefore, there is no solution to the given equation. It is an inconsistent equation.

d.

To determine

To solve the equation $-7x+5=2\left(x-10.1\right)$ and classify the equation based on the solution.

Answer to Problem 6CA

  $x=\frac{25.1}{9}=2.78$, conditional equation

Explanation of Solution

Given:

  $-7x+5=2\left(x-10.1\right)$

Concept used:

1. If an equation is true for all values of the variable, then it is an identity equation.
2. If an equation is true for only some values of the variable, then it is a conditional equation.
3. If an equation results in a false statement, then it is an inconsistent equation.

Calculation:

  $-7x+5=2\left(x-10.1\right)$

Solving the equation

  $\begin{array}{l}\Rightarrow -7x+5=2x-20.1\\ \Rightarrow -7x-2x=-20.1-5\\ \Rightarrow -9x=-25.1\\ \Rightarrow 9x=25.1\\ \Rightarrow x=\frac{25.1}{9}\\ \Rightarrow x=2.78\end{array}$

Now, classifying the equation on the basis of the solution obtained.

Since the solution set consists of one number: $\left\{2.78\right\}$ , it is the only solution and, therefore, the given equation is conditional.

Conclusion:

Therefore, the solution to the given equation is $x=2.78$ and it is a conditional equation.

e.

To determine

To solve the equation $6\left(2x+4\right)=4\left(4x+10\right)$ and classify the equation based on the solution.

Answer to Problem 6CA

  $x=-4$, conditional equation

Explanation of Solution

Given:

  $6\left(2x+4\right)=4\left(4x+10\right)$

Concept used:

1. If an equation is true for all values of the variable, then it is an identity equation.
2. If an equation is true for only some values of the variable, then it is a conditional equation.
3. If an equation results in a false statement, then it is an inconsistent equation.

Calculation:

  $6\left(2x+4\right)=4\left(4x+10\right)$

Solving the equation

  $\begin{array}{l}\Rightarrow 12x+24=16x+40\\ \Rightarrow 12x-16x=40-24\\ \Rightarrow -4x=16\\ \Rightarrow x=-\frac{16}{4}\\ \Rightarrow x=-4\end{array}$

Now, classifying the equation on the basis of the solution obtained.

Since the solution set consists of one number: $\left\{-4\right\}$ , it is the only solution and, therefore, the given equation is conditional.

Conclusion:

Therefore, the solution to the given equation is $x=-4$ and it is a conditional equation.

f

To determine

To solve the equation $8\left(3x+4\right)=2\left(12x+16\right)$ and classify the equation based on the solution.

Answer to Problem 6CA

  $x=$ all real numbers, Identity equation

Explanation of Solution

Given:

  $8\left(3x+4\right)=2\left(12x+16\right)$

Concept used:

1. If an equation is true for all values of the variable, then it is an identity equation.
2. If an equation is true for only some values of the variable, then it is a conditional equation.
3. If an equation results in a false statement, then it is an inconsistent equation.

Calculation:

  $8\left(3x+4\right)=2\left(12x+16\right)$

Solving the equation

  $\Rightarrow 24x+32=24x+32$

Now, classifying the equation on the basis of the solution obtained.

Since the solution set consists of all values that make the statement true. For this equation, the solution set is all real numbers because any real number substituted for $x$ will make the equation true. Therefore, the given equation is identical.

Conclusion:

Therefore, the solution set to the given equation is all real numbers and it is an identical equation.