II.

Any term, therefore, which occurs on both sides with the same sign may be removed from both without affecting the equality; and the sign of every term of an equation may be changed without affecting the equality.

54. Verification.

When the root is substituted for its symbol in the given equation, and the equation reduces to an identity, the root is said to be verified. We will illustrate by examples:

1. What number added to twice itself gives $24$?

Let $x$ stand for the number; then $2x$ will stand for twice the number, and the number added to twice itself will be $x+2x$.

But the number added to twice itself is $24$.

x + 2x = 24. lint Combine $x$ and $2x$, 3x = 24. Divide by $3$, the coefficient of $x$, x = 8. rint(Ax. 4)

Therefore the required number is $8$.

2. If $4x-5$ stands for $19$, for what number does $x$ stand?

We have the equation

4x - 5 = 19. lint Transpose $-5$, 4x = 19 + 5. lint Combine, 4x = 24. lint Divide by $4$, x = 6. rint(Ax. 4) [1] lint Verification. 4x - 5 = 19, 4 × 6 - 5 = 19, 24 - 5 = 19, 19 = 19.

3. If $3x-7$ stands for the same number as $14-4x$, what number does $x$ stand for?

We have the equation $3x-7=14-4x.$

Transpose $-4x$ to the left side, and $-7$ to the right side,

3x + 4x = 14 + 7. lint Combine, 7x = 21. lint Divide by $7$, x = 3. [1] lint Verification. 3x - 7 = 14 - 4x, 3 × 3 - 7 = 14 - 4 × 3, 2 = 2.

4. Solve the equation $7(x-1)-30=4(x-4).$ We have the equation $7(x-1)-30=4(x-4).$

Remove the parentheses,

7x - 7 - 30 = 4x - 16. lint Then 7x - 4x = 7 + 30 - 16. lint Combine, 3x = 21. lint Divide by $3$, x = 7. [1] lint Verification. 7(7 - 1) - 30 = 4(7 - 4), 7 × 6 - 30 = 4 x 3, 42 - 30 = 12, 12 = 12.

Exercise 9.

Find the number that $x$ stands for, if:

1. $3x=x+8$.