Exercise 85 Page 613

Practice makes perfect

a Before we begin, let's call the given equation

\frac{1}{2} (6 x - 14) + 5 x

=

2 - 3 x + 8

a basic equation. To decide which equations are equivalent to the basic equation, we will distribute

\frac{1}{2}

on the left-hand side to simplify it first.

\frac{1}{2} (6 x - 14) + 5 x = 2 - 3 x + 8 ⇕ 3 x - 7 + 5 x = 2 - 3 x + 8

Let's now compare the simplified form with the given equation from this part.

Basic equation	Given equation
$3 x - 7 + 5 x = 2 - 3 x + 8$	$3 x - 7 + 5 x = 10 - 3 x$

Notice that there are still some differences. Fortunately, in the simplified form of the basic equation we have like terms. Thus, we can combine them.

3 x - 7 + 5 x = 2 - 3 x + 8 ⇕ 3 x - 7 + 5 x = 10 - 3 x

We can see that now the simplified form of the basic equation is the same as the given equation from this part. Therefore, they are equivalent.

b Let's compare the given equation to the simplified form of the basic equation that we found in Part A.

Basic equation	Given equation
$3 x - 7 + 5 x = 2 - 3 x + 8$	$3 x - 14 + 5 x = 2 - 3 x + 8$

We can see that the above equations are the same except for one different term. This term cannot be rewritten in a way that will make the equations the same. Thus, the equations are not equivalent.

c Let's compare the given equation to the simplified form of the basic equation that we found in Part A.

Basic equation	Given equation
$3 x - 7 + 5 x = 2 - 3 x + 8$	$8 x - 14 = 10 - 3 x$

Notice that there are still some differences. Fortunately, in the simplified form of the basic equation we have like terms. Thus, we can combine them.

3 x - 7 + 5 x = 2 - 3 x + 8 ⇕ 8 x - 7 = 10 - 3 x

Let's compare the equations one more time using the obtained above form of the basic equation and the equation given in this part.

Basic equation	Given equation
$8 x - 7 = 10 - 3 x$	$8 x - 14 = 10 - 3 x$

We can see that the above equations are the same except for one different term. This term cannot be rewritten in a way that will make the equations be the same. Thus, the equations are not equivalent.

d Let's compare the given equation to the simplified form of the basic equation that we found in Part A.

Basic equation	Given equation
$3 x - 7 + 5 x = 2 - 3 x + 8$	$6 x - 14 + 10 = 4 - 6 x + 16$

Notice that there are significant differences. Every coefficient from the equation given in this part is greater than coefficients in the basic equation. Let's multiply both sides of the given equation by