Solve equations with variables on both sides, and apply linear equations to real word problems.
When a variable appears on both sides of an equation, collect all variable terms on one side and all constants on the other, then solve as usual.
Solve: 5x + 3 = 2x + 12
5x − 2x = 12 − 3 → 3x = 9 → x = 3
Solve: 7x − 4 = 3x + 16
7x − 3x = 16 + 4 → 4x = 20 → x = 5
Word problems can be translated into linear equations by assigning a variable to the unknown quantity, then writing an equation based on the relationships described.
The sum of two consecutive numbers is 45. Find the numbers.
Let the numbers be x and x+1. x + (x+1) = 45 → 2x = 44 → x = 22; the numbers are 22 and 23
A father is 3 times as old as his son. In 5 years, he will be 2.5 times as old. Find their current ages.
Let son's age = x, father's age = 3x. (3x+5) = 2.5(x+5) → 3x+5 = 2.5x+12.5 → 0.5x = 7.5 → x = 15; son is 15, father is 45
1. Solve: 6x + 5 = 3x + 20
2. Solve: 8x − 7 = 5x + 11
3. The sum of two consecutive even numbers is 74. Find the numbers.
4. Three times a number, decreased by 8, equals the number increased by 12. Find the number.
5. A mother is 4 times as old as her daughter. In 6 years, she will be 3 times as old. Find their current ages.