EXERCISE 2.1 - Step by Step Solutions
1. Solve: \(3x = 2x + 18\)
Transpose \(2x\) to the left:
\(3x - 2x = 18\)
\(x = 18\)
Check:
LHS: \(3x = 3 \times 18 = 54\)
RHS: \(2x + 18 = 36 + 18 = 54\)
LHS = RHS ✔️
2. Solve: \(5t - 3 = 3t - 5\)
Transpose \(3t\) to left, \(-3\) to right:
\(5t - 3t = -5 + 3\)
\(2t = -2\)
\(t = -1\)
Check:
LHS: \(5(-1) - 3 = -8\)
RHS: \(3(-1) - 5 = -8\)
LHS = RHS ✔️
3. Solve: \(5x + 9 = 5 + 3x\)
Transpose \(3x\) to left, \(9\) to right:
\(5x - 3x = 5 - 9\)
\(2x = -4\)
\(x = -2\)
Check:
LHS: \(5(-2) + 9 = -1\)
RHS: \(5 + 3(-2) = -1\)
LHS = RHS ✔️
4. Solve: \(4z + 3 = 6 + 2z\)
Transpose \(2z\) to left, \(3\) to right:
\(4z - 2z = 6 - 3\)
\(2z = 3\)
\(z = 1.5\)
Check:
LHS: \(4(1.5) + 3 = 9\)
RHS: \(6 + 2(1.5) = 9\)
LHS = RHS ✔️
5. Solve: \(2x - 1 = 14 - x\)
Transpose \(-x\) to left, \(-1\) to right:
\(2x + x = 14 + 1\)
\(3x = 15\)
\(x = 5\)
Check:
LHS: \(2(5) - 1 = 9\)
RHS: \(14 - 5 = 9\)
LHS = RHS ✔️
6. Solve: \(8x + 4 = 3(x - 1) + 7\)
Expand RHS:
\(8x + 4 = 3x - 3 + 7\)
Simplify RHS:
\(8x + 4 = 3x + 4\)
Transpose \(3x\) to left:
\(8x - 3x = 4 - 4\)
\(5x = 0\)
\(x = 0\)
Check:
LHS: \(8(0) + 4 = 4\)
RHS: \(3(0-1) + 7 = -3+7 = 4\)
LHS = RHS ✔️
7. Solve: \(\dfrac{4}{5}(x+10) = x\)
Expand:
\(\dfrac{4}{5}x + 8 = x\)
Transpose:
\(8 = x - \dfrac{4}{5}x\)
\(8 = x\left(1 - \dfrac{4}{5}\right) = x \times \dfrac{1}{5}\)
\(x = 8 \times 5 = 40\)
Check:
LHS: \(\dfrac{4}{5}(40+10) = \dfrac{4}{5}(50) = 40\)
RHS: \(x=40\)
LHS = RHS ✔️
8. Solve: \(\dfrac{2x}{3} + 1 = \dfrac{7x}{15} + 3\)
Transpose x terms to one side and constants to the other:
\(\dfrac{2x}{3} - \dfrac{7x}{15} = 2\)
By Cross Multiplication:
\(\dfrac{2x \times 15 - 7x \times 3}{3 \times 15} = 2\)
\(\dfrac{30x - 21x}{45} = 2 \;\Rightarrow\; \dfrac{9x}{45} = 2\)
\(\dfrac{x}{5} = 2\)
\(x=10\)
Check:
LHS: \(\dfrac{20}{3} + 1 = \dfrac{23}{3}\)
RHS: \(\dfrac{70}{15} + 3 = \dfrac{14}{3}+3 = \dfrac{23}{3}\)
LHS = RHS ✔️
9. Solve: \(2y + \dfrac{5}{3} = \dfrac{26}{3} - y\)
Transpose terms with y to one side and constants to the other:
\(2y + y = \dfrac{26}{3} - \dfrac{5}{3}\)
Simplify :
\(3y = \dfrac{26-5}{3}\)
\(3y = \dfrac{21}{3}\)
\(3y = 7\)
\(y = \dfrac{7}{3}\)
Check:
LHS: \(2\dfrac{7}{3} + \dfrac{5}{3} = \dfrac{19}{3}\)
RHS: \(\dfrac{26}{3} - \dfrac{7}{3} = \dfrac{19}{3}\)
LHS = RHS ✔️
10. Solve: \(3m = 5m - \dfrac{8}{5}\)
Transpose 5m to LHS and \(\frac{-8}{5}\) to RHS:
\(3m - 5m = -\dfrac{8}{5}\)
\(-2m = -\dfrac{8}{5}\)
Cancel - (negative sign) from both sides, equation becomes :
\(2m = \dfrac{8}{5}\)
Transpose 2 to RHS:
\(m = \dfrac{8}{5} \times \dfrac{1}{2}\)
\(m = \dfrac{4}{5}\)
Check:
LHS: \(3\times\dfrac{4}{5} = \dfrac{12}{5}\)
RHS: \(5\times\dfrac{4}{5} - \dfrac{8}{5} = 4 - \dfrac{8}{5} = \dfrac{12}{5}\)
LHS = RHS ✔️