Question

(\frac{4x + 1}{6}=\frac{x + 8}{5})

Explanation:

Step1: Cross - multiply to eliminate fractions

To solve the equation \(\frac{4x + 1}{6}=\frac{x + 8}{5}\), we cross - multiply. Cross - multiplying gives us \(5(4x + 1)=6(x + 8)\). The reason for cross - multiplying is that if \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d=b\times c\).

Step2: Expand both sides of the equation

Expanding the left - hand side: \(5\times(4x + 1)=5\times4x+5\times1 = 20x+5\).
Expanding the right - hand side: \(6\times(x + 8)=6\times x+6\times8=6x + 48\).
So our equation becomes \(20x+5 = 6x + 48\).

Step3: Subtract \(6x\) from both sides

Subtract \(6x\) from both sides of the equation \(20x+5 = 6x + 48\) to get \(20x-6x+5=6x - 6x+48\).
Simplifying, we have \(14x+5 = 48\).

Step4: Subtract 5 from both sides

Subtract 5 from both sides of the equation \(14x+5 = 48\): \(14x+5 - 5=48 - 5\).
Simplifying, we get \(14x=43\).

Step5: Divide both sides by 14

Divide both sides of the equation \(14x = 43\) by 14: \(x=\frac{43}{14}\) (or approximately \(x\approx3.07\)). Wait, let's check the calculation again.

Wait, let's re - do the steps:

Starting from \(\frac{4x + 1}{6}=\frac{x + 8}{5}\)

Cross - multiply: \(5(4x + 1)=6(x + 8)\)

Expand: \(20x+5 = 6x + 48\)

Subtract \(6x\) from both sides: \(20x-6x+5=6x - 6x + 48\)

\(14x+5 = 48\)

Subtract 5: \(14x=48 - 5=43\)

\(x=\frac{43}{14}\approx3.07\). Wait, but let's check if we made a mistake in cross - multiplying.

Wait, no, the cross - multiplication is correct. \(a/b=c/d\) implies \(a\times d=b\times c\). So \((4x + 1)\times5=6\times(x + 8)\)

\(20x+5 = 6x + 48\)

\(20x-6x=48 - 5\)

\(14x=43\)

\(x = \frac{43}{14}\)

Answer:

\(x=\frac{43}{14}\) (or \(x\approx3.07\))