【PEP Edition】Junior High Math, Grade 7, Volume 1: Secrets of the Balance: Basic Properties of Equations

An equation is like a precision balance in the world of mathematics. Solving an equation is essentially the art of "maintaining balance." Our goal is crystal clear: through legitimate operations, we gradually simplify the tangled algebraic expressions, until one side of the balance holds nothing but the lonely unknown $x$, and the other side reveals its true value.

The Two Fundamental Properties of Equations

To transform an equation without breaking its balance, we must follow two core rules:

Property 1 (Additive Conservation): Adding (or subtracting) the same number (or expression) to both sides of an equation keeps it balanced. This is like adding or removing equal weights from both pans of a balance, and it is commonly used to "eliminate" extra constant terms.
Property 2 (Multiplicative Conservation): Multiplying both sides of an equation by the same number, or dividing both sides by the same nonzero number, keeps it balanced. This is used to adjust the coefficient of the unknown, bringing it back to a clean 1.

Remember: solving an equation means gradually transforming it into the form $x = a$. Property 1 handles addition and subtraction, Property 2 handles multiplication and division, and the goal is always to reveal $x$ for what it truly is!

Key formulas: If $a=b$, then $a \pm c = b \pm c$; if $a=b$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c} (c \neq 0)$.

QUESTION 1

To solve the equation $x - 5 = 6$ using the properties of equations, the most appropriate first step is:

Subtract 5 from both sides

Add 5 to both sides

Multiply both sides by 5

Divide both sides by 6

QUESTION 2

Using the properties of equations to solve $0.3x = 45$, the value of $x$ is:

$13.5$

$15$

$150$

$1500$

QUESTION 3

To solve the equation $5x + 4 = 0$, what should you do?

Subtract 4 from both sides, then divide by 5

Add 4 to both sides, then divide by 5

Divide both sides by 5, then subtract 4

Multiply both sides by 5, then subtract 4

QUESTION 4

Using the properties of equations to solve $2 - \f\frac{1}{4}x = 3$, the solution is:

$x = 4$

$x = -4$

$x = 20$

$x = -20$

QUESTION 5

Write "5 more than $a$ equals 8" as an equation:

$a - 5 = 8$

$5a = 8$

$a + 5 = 8$

$a + 8 = 5$

QUESTION 6

Write "one third of $b$ equals 9" as an equation:

$\f\frac{1}{3}b = 9$

$3b = 9$

$b + \f\frac{1}{3} = 9$

$b - 3 = 9$

QUESTION 7

Write "the sum of twice $x$ and 10 equals 18" as an equation:

$2x - 10 = 18$

$x^2 + 10 = 18$

$2x + 10 = 18$

$2(x + 10) = 18$

QUESTION 8

Write "the difference of one third of $x$ minus $y$ equals 6" as an equation:

$\f\frac{1}{3}x - y = 6$

$\f\frac{1}{3}(x - y) = 6$

$3x - y = 6$

$x - \f\frac{1}{3}y = 6$

QUESTION 9

Tree-planting problem: If each person plants 10 trees, 6 trees are left over; if each person plants 12 trees, 6 trees are short. Let the number of people be $x$. The equation based on the equal total number of saplings is:

$10x - 6 = 12x + 6$

$10x + 6 = 12x - 6$

$\f\frac{x}{10} + 6 = \f\frac{x}{12} - 6$

$10(x + 6) = 12(x - 6)$

QUESTION 10

Mountain-climbing problem: Zhang Hua climbs at $10$ m/min and sets off 30 min earlier; Li Ming climbs at $15$ m/min. If they reach the summit at the same time, and Li Ming takes $t$ min, the equation should be:

$15t = 10(t - 30)$

$15t = 10(t + 30)$

$15(t + 30) = 10t$

$\f\frac{t}{15} = \f\frac{t + 30}{10}$