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Moving terms across the equals sign

Linear equations

Japanese school year: Junior high 1

What you learn

You will understand the principle of transposing terms by reversing their signs across the equals sign to solve linear equations. This operation serves as the foundation for solving various mathematical problems involving unknowns. Prior knowledge of the basic properties of equality and signed numbers is helpful.

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Key points

This transposition rule moves an added term +a+a across the equals sign. Its sign reverses, turning into subtraction −a-a.

x+a=b  ⟺  x=b−ax + a = b \iff x = b - a

This transposition rule moves a subtracted term −a-a across the equals sign. Its sign reverses, turning into addition +a+a.

x−a=b  ⟺  x=b+ax - a = b \iff x = b + a

This formula solves a linear equation for xx. Dividing both sides of ax=bax = b by the non-zero coefficient aa isolates the unknown to give x=bax = \frac{b}{a}.

ax=b  ⟺  x=ba(a≠0)ax = b \iff x = \frac{b}{a} \quad (a \neq 0)

This is the standard procedure for solving linear equations. Move terms with xx to the left and constants to the right, simplify into the form ax=bax = b, and then solve.

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