An equation is a statement that two expressions are equal, connected by the '=' sign. A simple equation has one unknown variable and the variable's power is 1. Solving an equation means finding the value of the variable that makes the statement true.
Setting up an Equation:
We translate word problems into equations using a variable (usually x or n) to represent the unknown. For example, "a number increased by 5 equals 12" becomes x + 5 = 12. Identifying the correct operation and the unknown is the first step in solving any word problem algebraically.
Solving an Equation — Balance Method:
Think of the equation as a balance scale. Whatever operation you apply to one side, you must apply to the other side to keep it balanced. This ensures the equality is preserved throughout every step of the solution.
Transposing:
To transpose a term means to move it to the other side of the equation with the opposite sign. Addition becomes subtraction and vice versa; multiplication becomes division and vice versa. Transposing is simply a shortcut for the balance method.
Checking the Solution:
Always substitute the found value back into the original equation to verify. If the left-hand side equals the right-hand side after substitution, the answer is correct. This step is essential and should never be skipped.
Solve 3x + 5 = 14.
Subtract 5 from both sides: 3x = 9. Divide both sides by 3: x = 3.
Check: 3(3) + 5 = 9 + 5 = 14. Correct.
Solve 2y - 7 = 11.
Add 7 to both sides: 2y = 18. Divide by 2: y = 9.
Solve (x/4) + 3 = 7.
Subtract 3: x/4 = 4. Multiply by 4: x = 16.
Solve 5(x - 2) = 20.
Divide both sides by 5: x - 2 = 4. Add 2: x = 6.
The sum of three consecutive integers is 48. Find them.
Let first integer = n. Then n + (n+1) + (n+2) = 48. 3n + 3 = 48. 3n = 45. n = 15. The integers are 15, 16, 17.
Raju is 3 years older than his sister. Together their ages sum to 23. Find their ages.
Let sister's age = x. Raju's age = x + 3. x + (x+3) = 23. 2x = 20. x = 10. Sister is 10, Raju is 13.
A number added to its double gives 36. Find the number.
Let number = n. n + 2n = 36. 3n = 36. n = 12.
- Key Formulas / Rules:
- If x + a = b, then x = b - a.
- If x - a = b, then x = b + a.
- If ax = b, then x = b/a.
- If x/a = b, then x = a x b.
Common mistakes
When transposing, always change the sign. For example, if you move +5 from the left side to the right, it becomes -5. Also, when multiplying or dividing, apply the operation to the ENTIRE side, not just one term.
Summary
A simple equation is solved by performing the same operation on both sides to isolate the variable. Transposing allows us to move terms across the equals sign with sign changes. Always verify by substituting the answer back.