Linear Equations in One Variable
ICSE Class 8 · Mathematics · Chapter 14
What this chapter covers
- Equations and their solutions
- The balance: properties of equality
- Transposition: a shortcut
- Equations with the variable on both sides
- Equations with brackets
- Equations with fractions and decimals
- Checking your answer and avoiding common errors
- From words to equations
- Solving word problems
Key points
- A linear equation in one variable has a single variable whose highest power is 1.
- The solution, or root, is the value of the variable that makes both sides equal.
- The same number may be added to or subtracted from both sides, and both sides may be multiplied or divided by the same non-zero number.
- Each such step gives an equivalent equation, so the solution never changes along the way.
- Transposition moves a term across the equals sign with the opposite operation; a multiplier becomes a divisor and keeps its sign.
- With the variable on both sides, collect variable terms on one side and constants on the other.
- Remove brackets with the distributive law, taking care with a minus sign in front.
- Clear fractions by multiplying every term by the LCM of the denominators, and clear decimals by multiplying by 10 or 100.
- For a word problem: name the unknown, form the equation, solve, check against the words, and answer with units.
- Always check the solution in the original equation.
Questions answered
Which of these are linear equations in one variable? (a) 3x - 4 = 11 (b) x2 = 9 (c) 2x + y = 5 (d) (p)/(5) = 2
(a) and (d). In (b) the power of x is 2, and (c) has two variables.
Is y = -2 a solution of 3y + 10 = 4?
Yes. LHS = 3 × (-2) + 10 = -6 + 10 = 4, which equals the RHS.
Solve 6x - 9 = 3x + 12.
6x - 3x = 12 + 9, so 3x = 21 and x = 7.
Solve 5(x - 2) = 3(x + 4).
5x - 10 = 3x + 12, so 2x = 22 and x = 11.
Solve (x)/(3) + (x)/(4) = 14.
Multiply by 12: 4x + 3x = 168, so 7x = 168 and x = 24.
Solve (2x - 1)/(3) = (x + 2)/(2).
Multiply both sides by 6: 2(2x - 1) = 3(x + 2), so 4x - 2 = 3x + 6 and x = 8. Check: 15 ÷ 3 = 5 and 10 ÷ 2 = 5.
Rohan solved 4x = -20 and wrote x = 5. What went wrong?
He changed the sign of the multiplier 4. Dividing -20 by 4 gives x = -5.
A number is multiplied by 5 and then 8 is subtracted. The result is 37. What is the number?
5x - 8 = 37, so 5x = 45 and the number is 9.