Verify
Verify solutions in Grade 9 algebra by substituting answers back into the original equation, and confirm that multiplying or dividing both sides by the same number preserves equation balance.
Key Concepts
Property Show that dividing both sides of the equation by 5 or multiplying both sides of the equation by $\frac{1}{5}$ will result in the same solution.
Examples Solve $\frac{2}{5}p = 7$ by dividing: $(\frac{2}{5} \div \frac{2}{5})p = (7 \div \frac{2}{5})$, which becomes $p = 7 \cdot \frac{5}{2} = \frac{35}{2}$. Solve $\frac{2}{5}p = 7$ by multiplying by the reciprocal: $(\frac{5}{2})\frac{2}{5}p = 7(\frac{5}{2})$, simplifying to $p = \frac{35}{2}$. For $ \frac{1}{2}x = 12$, you can multiply by $ 2$ to get $x = 24$. This is the same as dividing by $ \frac{1}{2}$!
Explanation Who needs one way when you can have two? When you see a fraction multiplied by a variable, you can either divide by that fraction or multiply by its reciprocal! Both methods are secret passages to the same answer. It's like having a math superpower to choose your own path to the solution.
Common Questions
Why is it important to verify your solution in algebra?
Verification catches arithmetic errors made during solving and confirms the answer actually satisfies the original equation. Always substitute your solution back into the original equation and check both sides are equal.
How do you verify that dividing both sides by 5 is equivalent to multiplying by 1/5?
Dividing by 5 means multiplying by its reciprocal, 1/5. Both operations are identical: 10 ÷ 5 = 2 and 10 × (1/5) = 2. This equivalence justifies either approach when isolating a variable coefficient.
How do you verify the solution x = 4 for the equation 3x - 2 = 10?
Substitute x = 4 into the original: 3(4) - 2 = 12 - 2 = 10. Since the left side equals 10 and the right side is 10, the equation is satisfied. The solution x = 4 is verified.