Linear Equations

An equation writes the unknown number as x and asks what value of x makes both sides equal. If 3 friends share candies equally and each gets 4, then x ÷ 3 = 4 tells us x is 12. Using the rule that an equation stays balanced when you add or multiply both sides by the same number, you isolate x and the answer appears. Here you balance the two sides on a scale and find the value of x for yourself.

An Equation is a Balance

a (coefficient of x)2

b (constant)3

c (right side)11

👀 Properties of Equality

①Adding the same number to both sides preserves equality

②Subtracting the same number preserves equality

③Multiplying both sides by the same number preserves equality

④Dividing both sides by the same nonzero number preserves equality

Solving Process

🔍 Solution Order

①Move constants to the right (transpose, sign flips)

②Compute the right side

③Divide both sides by the coefficient of x

④Write the answer as x = (number)

Why Transposing Works

Transposing

When a term moves across the equals sign, its sign flips

ax + b = c → ax = c - b (b moves to the right)

Check

Substitute the solution back into the original equation

Equal both sides ⇒ correct

💡 Transposing = Property of Equality

①Transposing is shorthand for "subtract the same value from both sides"

②From ax + b = c, subtract b: ax = c - b

③Net effect: b moves to the right with its sign flipped

Work It Out

Example 1

Solve the linear equation 2x + 3 = 11.

Transpose the constant 3 to the right.

2x = 11 − 3 = 8

Divide both sides by 2.

x = 8 ÷ 2 = 4

▸ x = 4

Watch the sign flip when transposing.

Example 2

Solve 3(x − 2) = x + 4.

Expand, then move x-terms left and constants right.

3x − 6 = x + 4 → 3x − x = 4 + 6

Simplify and divide by the coefficient.

2x = 10 → x = 5

▸ x = 5

With brackets, expand by the distributive law first, then transpose.

Exam Key Points

Solving Linear Equations

ax + b = c → x = c - ba

Transpose → simplify → divide by the coefficient

Grade-7 school exam type

The linear equation 2x + a = 5x − 7 has solution x = 3. What is the constant a?

①1

②2

③3

④4

⑤5

▸ ② 2

Substitute the solution x = 3 into the equation.

2 × 3 + a = 5 × 3 − 7

Simplify to find a.

6 + a = 8 ⇒ a = 2

🎯 Exam Key Points

①Must contain "=" to be an equation

②Linear: highest power of x is 1

③Always flip signs when transposing

④Multiply by the LCM to clear denominators

⑤Habit: verify by substitution

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Linear Expressions

Coordinate Plane & Graphs

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