Each equation of a pair such as 8x=4y+1 becomes a linear equation once both sides are written as powers of the same base. Method - Write every number as a power of one base, for example 8=23 and 3=3 12. - Combine the powers on each side with the laws of indices. - Equate the indices to get a linear equation; do the same with the other equation. - Solve the two linear equations together and check both original equations. Example Solve 8x=4y+1 and 27x=3y 3. 23x=22y+2 3x=2y+2 33x=3y+ 12 3x=y+ 12 Subtracting: 0=y+ 32, so y=- 32 and x=- 13. Check: 8- 13= 12=4- 12 and 27- 13= 13=3- 32 3. **Exam check** Equate indices only when each side is a single power of the same base. For a sum such as 2x+3y, let u=2x and v=3y instead.

Each equation of a pair such as \(8^x=4^{y+1}\) becomes a linear equation once both sides are written as powers of the same base.

Write every number as a power of one base, for example \(8=2^3\) and \(\sqrt3=3^{\frac12}\).

Combine the powers on each side with the laws of indices.

Equate the indices to get a linear equation; do the same with the other equation.

Solve the two linear equations together and check both original equations.

Solve \(8^x=4^{y+1}\) and \(27^x=3^y\sqrt3\).

\(\begin{aligned} 2^{3x}=2^{2y+2}\quad&\Rightarrow\quad 3x=2y+2 \\ 3^{3x}=3^{y+\frac12}\quad&\Rightarrow\quad 3x=y+\tfrac12 \end{aligned}\)

Subtracting: \(0=y+\tfrac32\), so \(y=-\tfrac32\) and \(x=-\tfrac13\).

Check: \(8^{-\frac13}=\tfrac12=4^{-\frac12}\) and \(27^{-\frac13}=\tfrac13=3^{-\frac32}\sqrt3\).

**Exam check** Equate indices only when each side is a single power of the same base. For a sum such as \(2^x+3^y\), let \(u=2^x\) and \(v=3^y\) instead.

Solving simultaneous exponential equations

Solving simultaneous exponential equations