Chapter 1.9, Problem 5LC

(a)

To determine

Solve the equation $ax=12$ for x. Assume $a\ne 0$

Answer to Problem 5LC

  $x=\frac{12}{a}$

Explanation of Solution

Given:

The equation $ax=12$ for x. Assume $a\ne 0$

Concept Used:

Addition or Subtraction Property of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}\text{+}\text{c}\text{=}\text{b}\text{+}\text{c}$ or If $\text{a}\text{=}\text{b}$ , then $\text{a}-\text{c}\text{=}\text{b}-\text{c}$

The property that states that if you add or subtract the same number to both sides of an equation, the sides remain equal (i.e., the equation continues to be true.)

Multiplication and Division Properties of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}·\text{c}=\text{b}·\text{c}$

If $\text{a}\text{=}\text{b}$ , then $\text{a}÷\text{c}=\text{b}÷\text{c}$ , where c is not equal to 0.

In other words, if two expressions are equal to each other and you multiply or divide (except for 0) the exact same constant to both sides, the two sides will remain equal. 

Calculation:

The equation: $ax=12$

Solve: $\begin{array}{l}\frac{ax}{a}=\frac{12}{a}[\text{divide}\text{each}\text{side}\text{by}a]\\ x=\frac{12}{a}\end{array}$

Thus, the solution of the equation $ax=12$ is $x=\frac{12}{a}$

(b)

To determine

Solve the equation $x+a=15$ for x. Assume $a\ne 0$

Answer to Problem 5LC

  $x=15-a$

Explanation of Solution

Given:

The equation: $x+a=15$; Assume $a\ne 0$

Concept Used:

Addition or Subtraction Property of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}\text{+}\text{c}\text{=}\text{b}\text{+}\text{c}$ or If $\text{a}\text{=}\text{b}$ , then $\text{a}-\text{c}\text{=}\text{b}-\text{c}$

The property that states that if you add or subtract the same number to both sides of an equation, the sides remain equal (i.e., the equation continues to be true.)

Multiplication and Division Properties of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}·\text{c}=\text{b}·\text{c}$

If $\text{a}\text{=}\text{b}$ , then $\text{a}÷\text{c}=\text{b}÷\text{c}$ , where c is not equal to 0.

In other words, if two expressions are equal to each other and you multiply or divide (except for 0) the exact same constant to both sides, the two sides will remain equal. 

Calculation:

The equation: $x+a=15$

Solve: $\begin{array}{l}x+a-a=15-a[\text{subtract}a\text{from}\text{both}\text{sides}]\\ x=15-a\end{array}$

Thus, the solution of the equation $x+a=15$is $x=15-a$

(c)

To determine

Solve the equation $-5=x-a$ for x. Assume $a\ne 0$

Answer to Problem 5LC

  $a-5$

Explanation of Solution

Given:

The equation $-5=x-a$ for x. Assume $a\ne 0$

Concept Used:

Addition or Subtraction Property of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}\text{+}\text{c}\text{=}\text{b}\text{+}\text{c}$ or If $\text{a}\text{=}\text{b}$ , then $\text{a}-\text{c}\text{=}\text{b}-\text{c}$

The property that states that if you add or subtract the same number to both sides of an equation, the sides remain equal (i.e., the equation continues to be true.)

Multiplication and Division Properties of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}·\text{c}=\text{b}·\text{c}$

If $\text{a}\text{=}\text{b}$ , then $\text{a}÷\text{c}=\text{b}÷\text{c}$ , where c is not equal to 0.

In other words, if two expressions are equal to each other and you multiply or divide (except for 0) the exact same constant to both sides, the two sides will remain equal. 

Calculation:

The equation: $-5=x-a$

Solve: $\begin{array}{l}-5+a=x-a+a[\text{add}a\text{to}\text{both}\text{sides}]\\ a-5=x\\ x=a-5\end{array}$

Thus, the solution of the equation $-5=x-a$ is $x=a-5$

(d)

To determine

Solve the equation $\frac{1}{a}x=10$ for x. Assume $a\ne 0$

Answer to Problem 5LC

  $10a$

Explanation of Solution

Given:

The equation $\frac{1}{a}x=10$ for x. Assume $a\ne 0$

Concept Used:

Addition or Subtraction Property of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}\text{+}\text{c}\text{=}\text{b}\text{+}\text{c}$ or If $\text{a}\text{=}\text{b}$ , then $\text{a}-\text{c}\text{=}\text{b}-\text{c}$

The property that states that if you add or subtract the same number to both sides of an equation, the sides remain equal (i.e., the equation continues to be true.)

Multiplication and Division Properties of Equality:

If $\text{a}\text{=}\text{b}$ , then $\text{a}·\text{c}=\text{b}·\text{c}$

If $\text{a}\text{=}\text{b}$ , then $\text{a}÷\text{c}=\text{b}÷\text{c}$ , where c is not equal to 0.

In other words, if two expressions are equal to each other and you multiply or divide (except for 0) the exact same constant to both sides, the two sides will remain equal. 

Calculation:

The equation: $\frac{1}{a}x=10$

Solve: $\begin{array}{l}a·\frac{1}{a}x=a·10[\text{Multiply}\text{each}\text{side}\text{by}a]\\ x=10a\end{array}$

Thus, the solution of the equation is $x=10a$